On Monotone Linear Relations and the Sum Problem in Banach Spaces by Liangjin Yao M.Sc., Yunnan University, 2006 M.Sc., The University of British Columbia, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The College of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) December 2011 c Liangjin Yao 2011 Abstract We study monotone operators in general Banach spaces. Properties and characterizations of monotone linear relations are presented. We focus on the “sum problem” which is the most famous open problem in Monotone Operator Theory, and we provide a powerful sufficient condition for the sum problem. We work on classical types of maximally monotone operators and provide affirmative answers to several open problems posed by Phelps and by Simons. Borwein-Wiersma decomposition and Asplund decomposition of maximally monotone operators are also studied. ii Preface My thesis is primarily based on the following twelve papers: [6–8] by Heinz H. Bauschke, Jonathan M. Borwein, Xiangfu Wang and Liangjin Yao; [14–18] by Heinz H. Bauschke, Xianfu Wang and Liangjin Yao; [88] by Xianfu Wang and Liangjin Yao; and [89–91] by Liangjin Yao. Specifically, the relationship between the above papers and my thesis is as follows: Chapter 3 is mainly based on the work in [15, 17, 18, 89]; Chapter 4 is mainly based on the work in [88]; Chapter 5 is mainly based on the work in [90, 91]; Chapter 6 is all based on the work in [6, 7]; Chapter 7 is mainly based on the work in [15, 17]; Chapter 8 is all based on the work in [8]; and Chapter 9 is mainly based on the work in [18]. For every multi-authored paper, each author contributed equally. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Preface List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Notation and examples . . . . . . . . . . . . . . . . . . . . . . 5 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1 3.1 Properties of linear relations . . . . . . . . . . . . . . . . . . 14 3.2 Properties of monotone linear relations . . . . . . . . . . . . 18 3.3 An unbounded skew operator on 2 (N) . . . . . . . . . . . . 30 3.4 The inverse Volterra operator on L2 [0, 1] . . . . . . . . . . . 42 iv Table of Contents 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Maximally monotone extensions of monotone linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1 . . . . . . . . . . . . . . 59 One linear relation: two equivalent formulations . . . 65 Auxiliary results on linear relations 4.1.1 4.2 Explicit maximally monotone extensions of monotone linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Minty parameterizations . . . . . . . . . . . . . . . . . . . . 4.4 Maximally monotone extensions with the same domain or the 66 79 same range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5 The sum problem . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1 Basic properties 5.2 Maximality of the sum of a (FPV) operator and a full domain operator 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Maximality of the sum of a linear relation and a subdifferential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.4 An example and comments . . . . . . . . . . . . . . . . . . . 135 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6 Classical types of maximally monotone operators 6.1 Introduction and auxiliary results . . . . . 139 . . . . . . . . . . . . . . . 139 v Table of Contents 6.2 Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type . . . . . . . . . . . . . . . . . . 142 6.3 The adjoint of a maximally monotone linear relation . . . . . 147 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7 Properties of monotone operators and the partial inf convolution of Fitzpatrick functions . . . . . . . . . . . . . . . . . 156 7.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.2 Fitzpatrick function of the sum of two linear relations . . . . 170 7.3 Fitzpatrick function of the sum of a linear relations and a normal cone operator 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . 178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8 BC–functions and examples of type (D) operators . . . . . 181 8.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.2 Main construction . . . . . . . . . . . . . . . . . . . . . . . . 185 8.3 Examples and applications 8.4 Discussion . . . . . . . . . . . . . . . . . . . 192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9 On Borwein-Wiersma decompositions of monotone linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 9.1 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.2 Uniqueness results . . . . . . . . . . . . . . . . . . . . . . . . 207 9.3 Characterizations and examples 9.4 When X is a Hilbert space 9.5 Discussion . . . . . . . . . . . . . . . . 214 . . . . . . . . . . . . . . . . . . . 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 vi Table of Contents 10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Appendices A Maple code Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 vii List of Figures 2.1 field plot of the linear operator A . . . . . . . . . . . . . . . . 10 viii List of Symbols A∗ the adjoint of a set-valued operator A p. 5 A−1 the inverse operator of A p. 5 A+ 1 2A + 21 A∗ p. 19 A◦ 1 2A − 21 A∗ p. 19 BX the closed unit ball of X z ∈ Z | z, d∗ = 0, D⊥ ∀d∗ ∈ D F p. 8 p. 5 p. 159 F1 1 F2 F1 2 F2 FA p. 181 the partial inf-convolution of F1 and F2 p. 156 Fitzpatrick function of A p. 6 F(z,z ∗ ) p. 141 H a Hilbert space p. 30 J the duality map p. 8 NC the normal cone operator of C p. 7 PC the projector on C p. 218 PX X × Y → X : (x, y) → x p. 8 PY X × Y → Y : (x, y) → y p. 8 QA p. 218 ix List of Symbols S–saturated p. 182 S⊥ z ∗ ∈ Z ∗ | z ∗ , s = 0, UX the open unit ball of X p. 8 X a real Banach space p. 5 IC the indicator mapping of C p. 7 Id identity mapping p. 8 ∀s ∈ S ΦA p. 5 p. 143 bdry C the boundary of C p. 7 conv C the convex hull of C p. 7 d(·, C) the distance function to a set C p. 7 dim F the dimension of F p. 7 dom A the domain of A p. 5 dom f f −1 (R) p. 7 2 (N) p. 30 gra A the graph of A p. 5 int C the interior of C p. 7 the weak∗ closure of C ∗ p. 7 the weak∗ closure of C ∗∗ p. 7 the norm closure of C p. 7 the weak closure of C p. 7 f the lower semicontinuous hull of f p. 7 ∂ε f the ε–subdifferential operator of f p. 8 ran A the range of A p. 5 C∗ w* C ∗∗ w* C C w Sgn p. 8 x List of Symbols σC the support function of C p. 7 f g the inf-convolution of f and g p. 8 f ⊕g f∗ p. 8 the Fenchel conjugate of f qA p. 7 p. 23 1 (N) p. 12 ιC the indicator function of C p. 7 ∂f the subdifferential operator of f p. 8 C −D {x − y | x ∈ C, y ∈ D} p. 7 xi Acknowledgements First, I would like to thank my supervisors Dr. Heinz Bauschke and Dr. Shawn Wang. Their kindness and enthusiasm for mathematics have deeply influenced me in many ways. Working with them has been a wonderful time in my life. I would also like to thank my committee member Dr. Yves Lucet for his help with my programs of study. I am very grateful to Dr. Stephen Simons, the external examiner, for his many valuable and constructive comments on my thesis. Finally, I thank all the good people in the mathematics department for their kind help, especially Ms. Pat Braham. xii Chapter 1 Introduction My thesis mainly focuses on monotone operators, which have proved to be a key class of objects in modern Optimization and Analysis. We start with linear relations, which are becoming a centre of attention in Monotone Operator Theory. In Chapter 3, we gather some basic properties about monotone linear relations, and conditions for them to be maximally monotone. We construct maximally monotone unbounded linear operators. We give some characterizations of the maximal monotonicity of linear operators and we also provide a brief proof of the Brezis-Browder Theorem. In Chapter 4, we focus on finding explicit maximally monotone linear subspace extensions of monotone linear relations, which generalize Crouzeix and Anaya’s recent work. The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. This is called the “sum problem”. The sum problem has an affirmative answer in reflexive spaces, but is still unsolved in general Banach spaces. In Chapter 5, we obtain a powerful sufficient condition for the sum problem to have an affirmative solution, which generalizes other well-known results for this 1 Chapter 1. Introduction problem obtained by different researchers in recent years. We also prove the case of the sum of a maximally monotone linear relation and the subdifferential operator. In Chapter 6, we study classical types of maximally monotone operators: dense type, negative-infimum type, Fitzpatrick-Phelps type, etc. We show that every maximally monotone operator of Fitzpatrick-Phelps type must be of dense type. We establish that for a maximally monotone linear relation, being of dense type, negative-infimum type, or Fitzpatrick-Phelps type is equivalent to the adjoint being monotone. The above results provide affirmative answers to two open problems: one posed by Phelps and Simons, and the other by Simons. The Fitzpatrick function is a very important tool in Monotone Operator Theory. In Chapter 7, we study the properties of the partial inf-convolution of the Fitzpatrick functions associated with maximally monotone operators. In Chapter 8, we construct some maximally monotone operators that are not of type (D). Using these operators, we show that the partial infconvolution of two BC-functions will not always be a BC-function, which provides a negative answer to a question posed by Simons. There are two well known decompositions of maximally monotone operators: Asplund Decomposition and Borwein-Wiersma Decomposition. In Chapter 9, we show that Borwein-Wiersma decomposability implies Asplund decomposability. We present characterizations of Borwein-Wiersma decomposability of maximally monotone linear relations in general Banach spaces and provide a more explicit decomposition in Hilbert spaces. In this thesis, we solve the following open problems. 2 Chapter 1. Introduction (1) Simons posed the following question in [74, page 199] concerning [72, Theorem 41.6] (See Corollary 5.3.6 or [16].): Let A : dom A → X ∗ be linear and maximally monotone, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Is A + NC necessarily maximally monotone? (2) Simons posed the following question in [74, Problem 47.6] (See Theorem 6.2.1 or [6].): Let A : dom A → X ∗ be linear and maximally monotone. Assume that A is of type (FP). Is A necessarily of type (NI)? (3) Simons posed the following question in [73, Problem 18, page 406] (See Corollary 6.2.2 or [7].): Let A : X ⇒ X ∗ be maximally monotone such that A is of type (FP). Is A necessarily of type (D)? (4) Phelps and Simons posed the following question in [63, Section 9, item 2] (See Corollary 6.3.3 or [6].): Let A : dom A → X ∗ be linear and maximally monotone. Assume that A∗ is monotone. Is A necessarily of type (D)? 3 Chapter 1. Introduction (5) Simons posed the following question in [74, Problem 22.12] (See Example 8.3.1(iii)&(v) or [8].): Let F1 , F2 : X × X ∗ → ]−∞, +∞] be proper lower semicontinuous and convex functions. Assume that F1 , F2 are BC– functions and that λ>0 λ [PX ∗ dom F1 − PX ∗ dom F2 ] is a closed subspace of X ∗ . Is F1 1 F2 necessarily a BC–function? The answers are yes, yes, yes, yes and no, respectively. 4 Chapter 2 Notation and examples In this chapter, we fix some notation and give some examples. Throughout this thesis, we assume that X is a real Banach space with norm · , that X ∗ is the continuous dual of X, and that X and X ∗ are paired by ·, · . Let A : X ⇒ X ∗ be a set-valued operator (also known as multifunction) from X to X ∗ , i.e., for every x ∈ X, Ax ⊆ X ∗ , and let gra A = (x, x∗ ) ∈ X × X ∗ | x∗ ∈ Ax be the graph of A. The inverse operator A−1 : X ∗ ⇒ X is given by gra A−1 = (x∗ , x) ∈ X ∗ × X | x∗ ∈ Ax ; the domain of A is dom A = x ∈ X | Ax = ∅ , and its range is ran A = A(X). If Z is a real Banach space with dual Z ∗ and a set S ⊆ Z, we define S ⊥ by S ⊥ = z ∗ , s = 0, z∗ ∈ Z ∗ | ∀s ∈ S . Given a subset D of Z ∗ , we define D⊥ [63] by D⊥ = z ∈ Z | z, d∗ = 0, ∀d∗ ∈ D . The adjoint of A, written A∗ , is defined by gra A∗ = (x∗∗ , x∗ ) ∈ X ∗∗ × X ∗ | (x∗ , −x∗∗ ) ∈ (gra A)⊥ = (x∗∗ , x∗ ) ∈ X ∗∗ × X ∗ | x∗ , a = a∗ , x∗∗ , ∀(a, a∗ ) ∈ gra A . See Example 2.1.2, Example 2.1.4, Section 3.3 and Cross’ book [38] for more information about linear relations. 5 Chapter 2. Notation and examples The Fitzpatrick function of A (see [45]) is given by FA : (x, x∗ ) ∈ X × X ∗ → sup (a,a∗ )∈gra A x, a∗ + a, x∗ − a, a∗ . (2.1) See Chapter 7 for more properties of the Fitzpatrick functions. Recall that A is monotone if ∀(x, x∗ ) ∈ gra A ∀(y, y ∗ ) ∈ gra A x − y, x∗ − y ∗ ≥ 0, (2.2) and maximally monotone if A is monotone and A has no proper monotone extension (in the sense of graph inclusion). We say (x, x∗ ) ∈ X × X ∗ is monotonically related to gra A if x − y, x∗ − y ∗ ≥ 0, ∀(y, y ∗ ) ∈ gra A. Let A : X ⇒ X ∗ be maximally monotone. We say A is of type FitzpatrickPhelps-Veronas (FPV) if for every open convex set U ⊆ X such that U ∩ dom A = ∅, the implication x ∈ U and (x, x∗ ) is monotonically related to gra A ∩ (U × X ∗ ) ⇒ (x, x∗ ) ∈ gra A holds. We say A is a linear relation if gra A is a linear subspace. Monotone operators have proven to be a key class of objects in modern Optimization and Analysis; see, e.g., [22–24], the books [9, 26, 33, 34, 48, 61, 68, 72, 74, 92, 93] and the references therein. We also adopt the standard notation 6 Chapter 2. Notation and examples used in these books: Given a subset C of X, int C is the interior of C, bdry C is the boundary of C, conv C is the convex hull of C, and C and C w are respectively the norm closure of C and weak closure of C. the set C ∗ ⊆ X ∗ , C ∗ w* For is the weak∗ closure of C ∗ . If C ∗∗ ⊆ X ∗∗ , C ∗∗ w* is the weak∗ closure of C ∗∗ in X ∗∗ with the topology induced by X ∗ . The indicator function of C, written as ιC , is defined at x ∈ X by ιC (x) = 0, ∞, if x ∈ C; (2.3) otherwise. The indicator mapping IC : X → X ∗ is defined by IC (x) = 0, ∅, if x ∈ C; (2.4) otherwise. The distance function to the set C, written as d(·, C), is defined by x → inf c∈C x − c . The support function of C, written as σC , is defined by σC (x∗ ) = supc∈C c, x∗ . If D ⊆ X, we set C − D = {x − y | x ∈ C, y ∈ D}. For every x ∈ X, the normal cone operator of C at x is defined by NC (x) = x∗ ∈ X ∗ | supc∈C c − x, x∗ ≤ 0 , if x ∈ C; and NC (x) = ∅, if x ∈ / C (see Example 2.1.5 for more information). For x, y ∈ X, we set [x, y] = {tx + (1 − t)y | 0 ≤ t ≤ 1}. Let dim F stand for the dimension of a subspace F of X. Given f : X → ]−∞, +∞], we set dom f = f −1 (R) and f ∗ : X ∗ → [−∞, +∞] : x∗ → supx∈X ( x, x∗ − f (x)) is the Fenchel conjugate of f . The lower semicontinuous hull of f is denoted by f . If f is convex and dom f = 7 Chapter 2. Notation and examples ∅, then ∂f : X ⇒ X ∗ : x → x∗ ∈ X ∗ | (∀y ∈ X) y − x, x∗ + f (x) ≤ f (y) is the subdifferential operator of f . the ε–subdifferential Note that NC = ∂ιC For ε ≥ 0, of f is defined by ∂ε f : X ⇒ X ∗ : x → x∗ ∈ X ∗ | (∀y ∈ X) y − x, x∗ + f (x) ≤ f (y) + ε . We have ∂f = ∂0 f . Let g : X → ]−∞, +∞]. The inf-convolution of f and g, f g, is defined by f g : x → inf [f (y) + g(x − y)] . y∈X Let J be the duality map, i.e., the subdifferential of the function 1 2 · 2. By [61, Example 2.26], Jx = x∗ ∈ X ∗ | x∗ , x = x∗ · x , with x∗ = x . (2.5) Let Id be the identity mapping from X to X. Let Y be a real Banach space. We also set PX : X × Y → X : (x, y) → x, and PY : X × Y → Y : (x, y) → y. Let f : X → ]−∞, +∞] and g : Y → ]−∞, +∞]. We define (f ⊕g) on X ×Y by (f ⊕ g)(x, y) = f (x) + g(y) for every (x, y) ∈ X × Y . The open unit ball in X is denoted by UX = closed unit ball in X is denoted by BX = x ∈ X | x < 1 , the x∈X | x ≤1 and N = {1, 2, 3, . . .}. Let Sgn be defined by 1, if ξ > 0; Sgn : R ⇒ R : ξ → [−1, 1] , if ξ = 0; −1, if ξ < 0. 8 2.1. Some examples Throughout, we shall identify X with its canonical image in the bidual space X ∗∗ . Furthermore, X × X ∗ and (X × X ∗ )∗ = X ∗ × X ∗∗ are likewise paired via (x, x∗ ), (y ∗ , y ∗∗ ) = x, y ∗ + x∗ , y ∗∗ , where (x, x∗ ) ∈ X × X ∗ and (y ∗ , y ∗∗ ) ∈ X ∗ ×X ∗∗ . Unless mentioned otherwise, the norm on X ×X ∗ , written as · 1, is defined by (x, x∗ ) 1 = x + x∗ for every (x, x∗ ) ∈ X × X ∗. 2.1 Some examples Now we give some examples of linear relations and their adjoints. See Example 2.1.1, Example 2.1.2 and Example 2.1.4. Example 2.1.1 Figure 2.1 is the graph of the linear operator: 0 −1 A= . 1 0 Example 2.1.2 (Borwein) (See [21, Example 3.1].) Let A : Rn ⇒ Rn be defined by Ax = Bx + V, ∅, if x ∈ S; otherwise, where B ∈ Rn×n , S and V are subspaces of Rn . Then ∗ A x= B T x + S ⊥ , ∅, if x ∈ V ⊥ ; otherwise. 9 2.1. Some examples Figure 2.1: field plot of the linear operator A That is, gra A = span{(s1 , Bs1 ), . . . , (sp , Bsp ), (0, v1 ), . . . , (0, vq )} gra A∗ = span{(v1 , B v1 ), . . . , (vp , B vp ), (0, s1 ), . . . , (0, sq )} where, (s1 , . . . , sp ), (v1 , . . . , vq ) are respectively the bases of S and V and (v1 , . . . , vp ), (s1 , . . . , sq ) are respectively the bases of V ⊥ and S ⊥ Remark 2.1.3 In Example 2.1.2, take S = Rn and V = 0, then A = B and A∗ = B T = AT . Let’s go to an explicit example of a monotone linear relation. 10 2.1. Some examples Example 2.1.4 Let A : R3 ⇒ R3 be defined by 4 1 −1 1 2 1 x + span e1 , Ax = −1 1 2 ∅, if x ∈ span{e2 }; otherwise, where e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1). Then and 4 1 −1 1 2 1 x + span{e1 , e3 }, A∗ x = −1 1 2 ∅, if x ∈ span{e2 , e3 }; otherwise, gra A = span{(0, e1 ), (e2 , e1 + 2e2 + e3 )} gra A∗ = span{(0, e1 ), (0, e3 ), (e2 , e1 + 2e2 + e3 ), (e3 , −e1 + e2 + 2e3 )}. The following is the explicit formula for the normal cone operator in 1 (N). Example 2.1.5 (Rockafellar) Suppose that X= 1 (N), with norm (xn )n∈N = n∈N |xn |, so that 11 2.1. Some examples X∗ = ∞ (N) with (x∗n )n∈N ∗ = supn∈N |x∗n | . The normal cone operator 1 (N), NBX is maximally monotone; furthermore, for every x ∈ 0 , NBX (x) = R+ · Sgn(xn ) ∅, if x < 1; n∈N , if x = 1; if x > 1. Proof. By Fact 5.1.2, NBX is maximally monotone. We now turn to the formula for the normal cone operator. Clearly, NBX (x) = {0} if x < 1, and NBX (x) = ∅ if x > 1. Now we suppose x = 1. Assume x∗ ∈ ∞ (N). Then x∗ ∈ NBX (x) ⇔ x∗ , y − x ≤ 0, ∀y ∈ BX ⇔ x∗ ⇔ x∗ ∗ ∗ ≤ x∗ , x = x∗ , x . (2.6) Clearly, K Sgn(xn ) ∞ ,x n=1 = K x = K = K Sgn(xn ) ∞ , n=1 ∗ ∀K ≥ 0. Thus, by (2.6), K · Sgn(xn ) ∞ n=1 | K ≥ 0 ⊆ NBX (x). ∗ ∗ Let x∗ ∈ NBX (x). Assume x∗ = (x∗n )∞ n=1 . If x = 0, then x ∈ K · Sgn(xn ) |x∗n | ≤ K, ∞ n=1 | K ≥ 0 . Now assume K := x∗ ∗ = 0. Thus, ∀n ∈ N. Let n ∈ N. Now we consider two cases: 12 2.1. Some examples Case 1: xn = 0. Clearly, x∗n ∈ K [−1, 1] = K Sgn(0). Case 2: xn = 0. We can suppose xn > 0. By (2.6), we have K = x∗n xn + i=n x∗i xi ≤ x∗n xn + sup |x∗j | · |xi | = x∗n xn + K(1 − xn ) i=n j∈N ≤ Kxn + K(1 − xn ) = K. Hence x∗n xn + K(1 − xn ) = Kxn + K(1 − xn ). Thus, x∗n = K. Then x∗ ∈ K · Sgn(xn ) ∞ . n=1 That is, NBX (x) ⊆ Hence NBX (x) = K · Sgn(xn ) K · Sgn(xn ) ∞ n=1 ∞ n=1 |K≥0 . |K≥0 . 13 Chapter 3 Linear relations This chapter is mainly based on [15, 17, 18] by Bauschke, Wang and Yao, and my work in [89]. We give some background material on linear relations, present some sufficient conditions for a linear relation to be monotone, and construct some examples of maximally monotone linear relations. Furthermore, we provide a brief proof of the Brezis-Browder Theorem on the characterization of the maximal monotonicity of linear relations. Recently, linear relations have become an interesting topic and are comprehensively studied in Monotone Operator Theory: see [3–5, 14–19, 28– 32, 63, 75, 80, 83, 87, 89, 91]. 3.1 Properties of linear relations In this section, we gather some basic properties about monotone linear relations, and conditions for them to be maximally monotone. These results are used frequently in the sequel. We start with properties for general linear relations. If A : X ⇒ X ∗ is a linear relation that is at most single-valued, then we will identify A with the corresponding linear operator from dom A to X ∗ and (abusing notation slightly) also write A : dom A → X ∗ . An analogous comment applies conversely to a linear single-valued operator A 14 3.1. Properties of linear relations with domain dom A, which we will identify with the corresponding at most single-valued linear relation from X to X ∗ . Fact 3.1.1 (See [58, Proposition 2.6.6(c)] or [69, Theorem 4.7 and Theorem 3.12]). Let C be a subspace of X, and D be a subspace of X ∗ . Then (C ⊥ )⊥ = C = C w and (D⊥ )⊥ = D w* . Fact 3.1.2 (Attouch-Brezis) (See [2, Theorem 1.1] or [74, Remark 15.2]). Let f, g : X → ]−∞, +∞] be proper lower semicontinuous convex functions. Assume that λ>0 λ [dom f − dom g] is a closed subspace of X. Then (f + g)∗ (z ∗ ) = min {f ∗ (y ∗ ) + g∗ (z ∗ − y ∗ )}, ∗ ∗ y ∈X ∀z ∗ ∈ X ∗ . (3.1) The following result appeared in Cross’ book [38]. We give new proofs. The proof of Proposition 3.1.3(ix) was borrowed from [18, Remark 2.2]. Proposition 3.1.3 Let A : X ⇒ X ∗ be a linear relation. Then the following hold. (i) A0 is a linear subspace of X ∗ . (ii) Ax = x∗ + A0, (iii) (∀(α, β) ∈ R2 ∀x∗ ∈ Ax. {(0, 0)}) (∀x, y ∈ dom A) A(αx + βy) = αAx + βAy. 15 3.1. Properties of linear relations (iv) (A∗ )−1 = (A−1 )∗ . (v) (∀x ∈ dom A∗ )(∀y ∈ dom A) A∗ x, y = x, Ay is a singleton. (vi) If X is reflexive and gra A is closed, then A∗∗ = A. (vii) (dom A)⊥ = A∗ 0 and dom A = (A∗ 0)⊥ . (viii) If gra A is closed, then (dom A∗ )⊥ = A0 and dom A∗ w* = (A0)⊥ . ¯ ⊥ and thus dom A∗ is (weak∗ ) (ix) If dom A is closed, then dom A∗ = (A0) closed, where A¯ is the linear relation whose graph is the closure of the graph of A. (x) If k ∈ R {0}, then (kA)∗ = kA∗ . Proof. (i): Since gra A is a linear subspace, {0} × A0 = gra A ∩ {0} × X ∗ is a linear subspace and hence A0 is a linear subspace. (ii): Let x ∈ dom A and x∗ ∈ Ax. Then (x, x∗ +A0) = (x, x∗ )+(0, A0) ⊆ gra A and hence x∗ + A0 ⊆ Ax. On the other hand, let y ∗ ∈ Ax. We have (0, y ∗ − x∗ ) = (x, y ∗ ) − (x, x∗ ) ∈ gra A. Then y ∗ − x∗ ∈ A0 and thus y ∗ ∈ x∗ + A0. Hence Ax ⊆ x∗ + A0 and thus Ax = x∗ + A0. (iii): Let (α, β) ∈ R2 {(0, 0)} and {x, y} ⊆ dom A. We can suppose α = 0 and β = 0. Take x∗ ∈ Ax and y ∗ ∈ Ay. Since gra A is a linear subspace, αx∗ + βy ∗ ∈ A(αx + βy). By (ii), A(αx + βy) = αx∗ + βy ∗ + A0 = αx∗ + A0 + βy ∗ + A0 = α(x∗ + α1 A0) + β(y ∗ + β1 A0) = αAx + βAy. (iv): We have (x∗ , x∗∗ ) ∈ gra(A∗ )−1 ⇔ (x∗∗ , x∗ ) ∈ gra A∗ ⇔ (x∗ , −x∗∗ ) ∈ (gra A)⊥ ⇔ (x∗∗ , −x∗ ) ∈ (gra A−1 )⊥ ⇔ (x∗ , x∗∗ ) ∈ gra(A−1 )∗ . 16 3.1. Properties of linear relations (v): Let x ∈ dom A∗ and y ∈ dom A. Take x∗ ∈ A∗ x and y ∗ ∈ Ay. We have x∗ , y = y ∗ , x , ∀x∗ ∈ A∗ x, y ∗ ∈ Ay. Hence A∗ x, y and Ay, x are singleton and equal. (vi): We have (x, x∗ ) ∈ gra A∗∗ ⇔ (x∗ , −x) ∈ (gra A∗ )⊥ = ((gra −A−1 )⊥ )⊥ = gra −A−1 ⇔ (x, x∗ ) ∈ gra A. (vii): Clearly, (dom A)⊥ ⊆ A∗ 0. Let x∗ ∈ A∗ 0. We have x∗ , y + 0, Ay = 0, ∀y ∈ dom A. Then we have x∗ ∈ (dom A)⊥ and thus A∗ 0 ⊆ (dom A)⊥ . Hence (dom A)⊥ = A∗ 0. By Fact 3.1.1, dom A = (A∗ 0)⊥ . (viii): By Fact 3.1.1, x∗ ∈ A0 ⇔ (0, x∗ ) ∈ gra A = (gra A)⊥ ⇔ x∗ , y ∗∗ = 0, ⊥ = gra −(A∗ )−1 ⊥ ∀y ∗∗ ∈ dom A∗ ⇔ x∗ ∈ (dom A∗ )⊥ . Hence (dom A∗ )⊥ = A0. Take Y = X ∗ , by Fact 3.1.1 again, dom A∗ w* = (A0)⊥ . (ix): Let A¯ be the linear relation whose graph is the closure of the graph of A. Then dom A = dom A¯ and A∗ = A¯∗ . Then by Fact 3.1.2, ∗ ιX ∗ ×(A0) ¯ ⊥ = ι{0}×A0 ¯ + ι{0}×X ∗ ¯ = ιgra A ∗ = ιgra(−A¯∗ )−1 ιX ∗ ×{0} = ιX ∗ ×dom A¯∗ . ¯ ⊥ is closed. It is clear that dom A∗ = dom A¯∗ = (A0) (x): Let k ∈ R {0}. Then (x∗∗ , x∗ ) ∈ gra(kA)∗ ⇔ (x∗ , −x∗∗ ) ∈ (gra kA)⊥ ⇔ (x∗ , −kx∗∗ ) ∈ (gra A)⊥ ⇔ ( k1 x∗ , −x∗∗ ) ∈ (gra A)⊥ ⇔ (x∗∗ , k1 x∗ ) ∈ gra A∗ . Hence (kA)∗ = kA∗ . 17 3.2. Properties of monotone linear relations 3.2 Properties of monotone linear relations Proposition 3.2.1, Proposition 3.2.2 and Proposition 3.2.7 were established in reflexive spaces by Bauschke, Wang and Yao in [15, Proposition 2.2]. Here, we adapt the proofs to a general Banach space. Proposition 3.2.1 Let A : X ⇒ X ∗ be a linear relation. Then the following hold. (i) Suppose A is monotone. Then dom A ⊆ (A0)⊥ and A0 ⊆ (dom A)⊥ ; consequently, if gra A is closed, then dom A ⊆ dom A∗ w* ∩ X and A0 ⊆ A∗ 0. (ii) (∀x ∈ dom A)(∀z ∈ (A0)⊥ ) z, Ax is single-valued. (iii) (∀z ∈ (A0)⊥ ) dom A → R : y → z, Ay is linear. (iv) A is monotone ⇔ (∀x ∈ dom A) x, Ax is single-valued and x, Ax ≥ 0. (v) If (x, x∗ ) ∈ (dom A) × X ∗ is monotonically related to gra A and x∗0 ∈ Ax, then x∗ − x∗0 ∈ (dom A)⊥ . Proof. (i): Pick x ∈ dom A. Then there exists x∗ ∈ X ∗ such that (x, x∗ ) ∈ gra A. By the monotonicity of A and since (0, A0) ⊆ gra A, we have x, x∗ ≥ sup x, A0 . Since A0 is a linear subspace (Proposition 3.1.3(i)), we obtain x⊥A0. This implies dom A ⊆ (A0)⊥ and A0 ⊆ (dom A)⊥ . If gra A is closed, then Proposition 3.1.3(viii)&(vii) yield dom A ⊆ (A0)⊥ ⊆ (A0)⊥ = dom A∗ w* and A0 ⊆ A∗ 0. 18 3.2. Properties of monotone linear relations (ii): Take x ∈ dom A, x∗ ∈ Ax, and z ∈ (A0)⊥ . By Proposition 3.1.3(ii), z, Ax = z, x∗ + A0 = z, x∗ . (iii): Take z ∈ (A0)⊥ . By (ii), (∀y ∈ dom A) z, Ay is single-valued. Now let x, y be in dom A, and let α, β be in R. If (α, β) = (0, 0), then z, A(αx + βy) = z, A0 = 0 = α z, Ax + β z, Ay . And if (α, β) = (0, 0), then Proposition 3.1.3(iii) yields z, A(αx + βy) = z, αAx + βAy = α z, Ax + β z, Ay . This verifies linearity. (iv): “⇒”: This follows from (i), (ii), and the fact that (0, 0) ∈ gra A. “⇐”: If x and y belong to dom A, then Proposition 3.1.3(iii) yields x − y, Ax − Ay = x − y, A(x − y) ≥ 0. (v): Let (x, x∗ ) ∈ (dom A) × X ∗ be monotonically related to gra A, and take x∗0 ∈ Ax. For every (v, v ∗ ) ∈ gra A, we have that x∗0 + v ∗ ∈ A(x + v) (by Proposition 3.1.3(iii)); hence, x − (x + v), x∗ − (x∗0 + v ∗ ) ≥ 0 and thus v, v ∗ ≥ v, x∗ − x∗0 . Now take λ > 0 and replace (v, v ∗ ) in the last inequality by (λv, λv ∗ ). Then divide by λ and let λ → 0+ to see that 0 ≥ sup dom A, x∗ − x∗0 . Since dom A is linear, it follows that x∗ − x∗0 ∈ (dom A)⊥ . We say that a linear relation A : X ⇒ X ∗ is skew if gra A ⊆ gra(−A∗ ), equivalently, if x, x∗ = 0, ∀(x, x∗ ) ∈ gra A; furthermore, A is symmetric if gra A ⊆ gra A∗ ; equivalently, if x, y ∗ = y, x∗ , ∀(x, x∗ ), (y, y ∗ ) ∈ gra A. We define the symmetric part and the skew part of A via A+ = 12 A + 21 A∗ and A◦ = 12 A − 21 A∗ , (3.2) respectively. It is easy to check that A+ is symmetric and that A◦ is skew. 19 3.2. Properties of monotone linear relations Proposition 3.2.2 Let A : X ⇒ X ∗ be a monotone linear relation. Then the following hold. (i) If A is maximally monotone, then (dom A)⊥ = A0 and hence dom A = (A0)⊥ . (ii) If dom A is closed, then: A is maximally monotone ⇔ (dom A)⊥ = A0. (iii) If A is maximally monotone, then dom A∗ w* ∩ X = dom A = (A0)⊥ and A0 = A∗ 0 = A+ 0 = A◦ 0 = (dom A)⊥ . (iv) If A is maximally monotone and dom A is closed, then dom A∗ ∩ X = dom A. (v) If A is maximally monotone and dom A ⊆ dom A∗ , then A = A+ +A◦ , A+ = A − A◦ , and A◦ = A − A+ . Proof. (i): Since A + Ndom A = A + (dom A)⊥ is a monotone extension of A and A is maximally monotone, we must have A + (dom A)⊥ = A. Then A0 + (dom A)⊥ = A0. As 0 ∈ A0, (dom A)⊥ ⊆ A0. The reverse inclusion follows from Proposition 3.2.1(i). Then we have (dom A)⊥ = A0. By Fact 3.1.1, dom A = (A0)⊥ . (ii): “⇒”: This follows directly from (i). “⇐”: By our assumptions and Fact 3.1.1, dom A = (A0)⊥ . Let (x, x∗ ) be monotonically related to gra A. We have inf [ x − 0, x∗ − A0 ] ≥ 0. Then we have x ∈ (A0)⊥ and hence x ∈ dom A. Then by Proposition 3.2.1(v) and Proposition 3.1.3(ii), x∗ ∈ Ax. Hence A is maximally monotone. (iii): By (i) and Proposition 3.1.3(vii), A0 = (dom A)⊥ = A∗ 0 and thus A+ 0 = A◦ 0 = A0 = (dom A)⊥ . Then by Proposition 3.1.3(viii) and (i), 20 3.2. Properties of monotone linear relations dom A∗ w* ∩ X = (A0)⊥ = dom A. (iv): Apply (iii) and Proposition 3.1.3(ix) directly. (v): We show only the proof of A = A+ + A◦ as the other two proofs are analogous. Clearly, dom A+ = dom A◦ = dom A ∩ dom A∗ = dom A. Let x ∈ dom A, and x∗ ∈ Ax and y ∗ ∈ A∗ x. We write x∗ = x∗ +y ∗ 2 + x∗ −y ∗ 2 ∈ (A+ + A◦ )x. Then, by (iii) and Proposition 3.1.3(ii), Ax = x∗ + A0 = x∗ + (A+ + A◦ )0 = (A+ + A◦ )x. Therefore, A = A+ + A◦ . Corollary 3.2.3 below first appeared in [63, Corollary 2.6 and Proposition 3.2(h)] by Phelps and Simons. Voisei and Z˘ alinescu showed that the maximality part also holds in locally convex spaces [87, Proposition 23]. Corollary 3.2.3 Let A : X → X ∗ be monotone and linear. Then A is maximally monotone and continuous. Proof. By Proposition 3.2.2(ii), A is maximally monotone and thus gra A is closed. By the Closed Graph Theorem, A is continuous. Proposition 3.2.2(ii) provides a characterization of maximal monotonicity for certain monotone linear relations. More can be said in finitedimensional spaces. We require the following lemma, where dim F stands for the dimension of a subspace F of X. Lemma 3.2.4 and Proposition 3.2.5 were established by Bauschke, Wang and Yao in [18]. Lemma 3.2.4 Suppose that X is finite-dimensional and let A : X ⇒ X ∗ be a linear relation. Then dim(gra A) = dim(dom A) + dim A0. Proof. We shall construct a basis of gra A. By Proposition 3.1.3(i), A0 is a linear subspace. Let {x∗1 , . . . , x∗k } be a basis of A0, and let {xk+1 , . . . , xl } be 21 3.2. Properties of monotone linear relations a basis of dom A. From Proposition 3.1.3(ii), it is easy to show {(0, x∗1 ), . . . , (0, x∗k ), (xk+1 , x∗k+1 ), . . . , (xl , x∗l )} is a basis of gra A, where x∗i ∈ Axi , i ∈ {k + 1, . . . , l}. Thus dim(gra A) = l = dim(dom A) + dim A0. Lemma 3.2.4 allows us to get a satisfactory characterization of maximal monotonicities of linear relations in finite-dimensional spaces. Proposition 3.2.5 Suppose that X is finite-dimensional, set n = dim X, and let A : X ⇒ X ∗ be a monotone linear relation. Then A is maximally monotone if and only if dim gra A = n. Proof. Since linear subspaces of X are closed, we see from Proposition 3.2.2(ii) that A is maximally monotone ⇔ dom A = (A0)⊥ . (3.3) Assume first that A is maximally monotone. Then dom A = (A0)⊥ . By Lemma 3.2.4, dim(gra A) = dim(dom A)+dim(A0) = dim((A0)⊥ )+dim(A0) = n. Conversely, let dim(gra A) = n. By Lemma 3.2.4, we have that dim(dom A) = n − dim(A0). As dim((A0)⊥ ) = n − dim(A0) and dom A ⊆ (A0)⊥ by Proposition 3.2.1(i), we have that dom A = (A0)⊥ . By (3.3), A is maximally monotone. Next, we obtain a key criteria on concerning maximally monotone linear relations, which I will frequently use to construct maximally monotone linear subspace extensions in Chapter 4. Corollary 3.2.6 Let A : Rn ⇒ Rn be a monotone linear relation. The following are equivalent: (i) A is maximally monotone. 22 3.2. Properties of monotone linear relations (ii) dim gra A = n. (iii) dom A = (A0)⊥ . For a monotone linear relation A : X ⇒ X ∗ it will be convenient to define (as in, e.g., [5]) (∀x ∈ X) qA (x) = 1 x, Ax , 2 ∞, if x ∈ dom A; otherwise. Proposition 3.2.7 Let A : X ⇒ X ∗ be a monotone linear relation, let x and y be in dom A, and let λ ∈ R. Then qA is single-valued and λqA (x) + (1 − λ)qA (y) − qA (λx + (1 − λ)y) = λ(1 − λ)qA (x − y) = 12 λ(1 − λ) x − y, Ax − Ay . (3.4) Moreover, qA is convex. Proof. Proposition 3.2.1(iv) shows that qA is single-valued on dom A and that qA ≥ 0. Combining with Proposition 3.2.1(i)&(iii), we obtain (3.4). Then by (3.4), qA is convex. Fact 3.2.8 (Simons) (See [74, Lemma 19.7 and Section 22].) Let A : X ⇒ X ∗ be a monotone operator with convex graph such that gra A = ∅. Then the function g : X × X ∗ → ]−∞, +∞] : (x, x∗ ) → x, x∗ + ιgra A (x, x∗ ) (3.5) 23 3.2. Properties of monotone linear relations is proper and convex. Proof. It is clear that g is proper because gra A = ∅. To see that g is convex, let (a, a∗ ) and (b, b∗ ) be in gra A, and let λ ∈ ]0, 1[. Set µ = 1− λ ∈ ]0, 1[ and observe that λ(a, a∗ ) + µ(b, b∗ ) = (λa + µb, λa∗ + µb∗ ) ∈ gra A by convexity of gra A. Since A is monotone, it follows that λg(a, a∗ ) + µg(b, b∗ ) − g λ(a, a∗ ) + µ(b, b∗ ) = λ a, a∗ + µ b, b∗ − λa + µb, λa∗ + µb∗ = λµ a − b, a∗ − b∗ ≥ 0. Therefore, g is convex. Phelps and Simons proved Fact 3.2.9 in the unbounded linear case in [63, Proposition 3.2(a)], but their proof can also be adapted to a linear relation. For readers’ convenience, we write down their proof. Fact 3.2.9 (Phelps-Simons) Let A : X ⇒ X ∗ be a monotone linear relation. Then (x, x∗ ) ∈ X × X ∗ is monotonically related to gra A if and only if x, x∗ ≥ 0 and [ y ∗ , x + x∗ , y ]2 ≤ 4 x∗ , x y ∗ , y , ∀(y, y ∗ ) ∈ gra A. Proof. We have (x, x∗ ) ∈ X × X ∗ is monotonically related to gra A 24 3.2. Properties of monotone linear relations ⇔ λ2 y, y ∗ − λ [ y ∗ , x + x∗ , y ] + x, x∗ = λy ∗ − x∗ , λy − x ≥ 0, ∀λ ∈ R, ∀(y, y ∗ ) ∈ gra A ⇔ x, x∗ ≥ 0 and [ y ∗ , x + x∗ , y ]2 ≤ 4 x∗ , x y ∗ , y , ∀(y, y ∗ ) ∈ gra A (by [63, Lemma 2.1]). The proof of Proposition 3.2.10(iii) was borrowed from [30, Theorem 2]. Results very similar to Proposition 3.2.10(i)&(ii) are established in [89, Proposition 18.9]. Proposition 3.2.10 Let A : X ⇒ X ∗ be a monotone linear relation. Then (i) A+ is monotone, and qA + ιdom A+ = qA+ and thus qA+ is convex. (ii) gra A+ ⊆ gra ∂qA . If A+ is maximally monotone, then A+ = ∂qA . (iii) If A is maximally monotone, then A∗ |X is monotone. (iv) If A is maximally monotone and dom A is closed, then A∗ |X is maximally monotone. Proof. Let x ∈ dom A+ . (i): Since A is monotone, by Proposition 3.1.3(v) and Proposition 3.2.1(iv), qA+ = qA |dom A+ and A+ is monotone. Then by Proposition 3.2.7, qA+ is convex. Let y ∈ dom A. Then by Proposition 3.1.3(v) again, 0≤ 1 2 Ax − Ay, x − y = 1 2 Ay, y + 1 2 Ax, x − A+ x, y , (3.6) 25 3.2. Properties of monotone linear relations we have qA (y) ≥ A+ x, y − qA (x). Take lower semicontinuous hull at y and then deduce that qA (y) ≥ A+ x, y − qA (x). For y = x, we have qA (x) ≥ qA (x). On the other hand, qA (x) ≤ qA (x). Altogether, qA (x) = qA (x) = qA+ (x). Thus (i) holds. (ii): Let y ∈ dom A. By (3.6) and (i), qA (y) ≥ qA (x) + A+ x, y − x = qA (x) + A+ x, y − x . (3.7) Since dom qA ⊆ dom qA = dom A, by (3.7), qA (z) ≥ qA (x) + A+ x, z − ∀z ∈ dom qA . Hence A+ x ⊆ ∂qA (x). If A+ is maximally monotone, x, then A+ = ∂qA . Thus (ii) holds. (iii): Suppose to the contrary that A∗ |X is not monotone. By Proposition 3.2.1(iv), there exists (x0 , x∗0 ) ∈ gra A∗ with x0 ∈ X such that x0 , x∗0 < 0. Now we have −x0 − y, x∗0 − y ∗ = −x0 , x∗0 + y, y ∗ + x0 , y ∗ + −y, x∗0 = −x0 , x∗0 + y, y ∗ > 0, ∀(y, y ∗ ) ∈ gra A. (3.8) Thus, (−x0 , x∗0 ) is monotonically related to gra A. By the maximal monotonicity of A, (−x0 , x∗0 ) ∈ gra A. Then −x0 − (−x0 ), x∗0 − x∗0 = 0, which contradicts (3.8). Hence A∗ |X is monotone. (iv): By Proposition 3.1.3(ix), dom A∗ |X = (A0)⊥ and thus dom A∗ |X is closed. By Fact 3.1.1 and Proposition 3.2.2(i), (dom A∗ |X )⊥ = ((A0)⊥ )⊥ = A0 w∗ = A0. Then by Proposition 3.2.2(iii), (dom A∗ |X )⊥ = A∗ 0. Apply (iii) and Proposition 3.2.2(ii), A∗ |X is maximally monotone. 26 3.2. Properties of monotone linear relations Proposition 3.2.11 Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then A is symmetric ⇔ A = A∗ |X . Proof. “⇒”: Assume that A is symmetric, i.e., gra A ⊆ gra A∗ . Since A is maximally monotone, by Proposition 3.2.10(iii), A = A∗ |X . “⇐”: Obvious. Fact 3.2.12 (Phelps-Simons) (See [63, Theorem 2.5 and Lemma 4.4].) Let A : dom A → X ∗ be monotone and linear. The following hold. (i) If A is maximally monotone, then dom A is dense (and hence A∗ is at most single-valued). (ii) Assume that A is skew such that dom A is dense. Then dom A ⊆ dom A∗ and A∗ |dom A = −A. Fact 3.2.13 (Brezis-Browder) (See [30, Theorem 2].) Assume X is reflexive. Let A : X ⇒ X ∗ be a monotone linear relation such that gra A is closed. Then the following are equivalent. (i) A is maximally monotone. (ii) A∗ is maximally monotone. (iii) A∗ is monotone. In Theorem 3.2.15, established in [89, Theorem 18.5], we provide a new and simpler proof to show the hard part (iii)⇒(i) in Fact 3.2.13. We first need the following fact. 27 3.2. Properties of monotone linear relations Fact 3.2.14 (Simons-Z˘ alinescu) (See [77, Theorem 1.2] or [72, Theorem 10.6].) Assume X is reflexive. Let A : X ⇒ X ∗ be monotone. Then A is maximally monotone if and only if gra A + gra(−J) = X × X ∗ . Now we come to the hard part (iii)⇒(i) in Theorem 3.2.13. The proof was inspired by that of [93, Theorem 32.L]. Theorem 3.2.15 Assume X is reflexive. Let A : X ⇒ X ∗ be a monotone linear relation with closed graph. Suppose A∗ is monotone. Then A is maximally monotone. Proof. By Fact 3.2.14, it suffices to show that X × X ∗ ⊆ gra A + gra(−J). For this, let (x, x∗ ) ∈ X × X ∗ and we define g : X × X ∗ → ]−∞, +∞] by (y, y ∗ ) → 1 2 y∗ 2 + 1 2 y 2 + y ∗ , y + ιgra A (y − x, y ∗ − x∗ ). We have f : (y, y ∗ ) → y ∗ , y +ιgra A (y−x, y ∗ −x∗ ) = y ∗ , y +ιgra A+(x,x∗ ) (y, y ∗ ). By Fact 3.2.8 and the assumption that gra A is closed, f is proper lower semicontinuous and convex. Hence g is lower semicontinuous convex and coercive. According to [92, Theorem 2.5.1(ii)], g has minimizers. Suppose that (z, z ∗ ) is a minimizer of g. Then (z − x, z ∗ − x∗ ) ∈ gra A, hence, (x, x∗ ) ∈ gra A + (z, z ∗ ). (3.9) 28 3.2. Properties of monotone linear relations On the other hand, since (z, z ∗ ) is a minimizer of g, (0, 0) ∈ ∂g(z, z ∗ ). By a result of Rockafellar (see [37, Theorem 2.9.8] and [92, Theorem 3.2.4(ii)] or [60, Theorem 1.93 and Proposition 1.107(ii)]), there exist (z0∗ , z0 ) ∈ ∂(ιgra A (· − x, · − x∗ ))(z, z ∗ ) = ∂ιgra A (z − x, z ∗ − x∗ ) = (gra A)⊥ , and (v, v ∗ ) ∈ X × X ∗ with v ∗ ∈ Jz, z ∗ ∈ Jv such that (0, 0) = (z ∗ , z) + (v ∗ , v) + (z0∗ , z0 ). Then − (z + v), z ∗ + v ∗ ∈ gra A∗ . Since A∗ is monotone, z ∗ + v ∗ , z + v = z ∗ , z + z ∗ , v + v ∗ , z + v ∗ , v ≤ 0. Note that since z ∗ , v = z ∗ 2 = v 2 , v∗ , z = v∗ 2 (3.10) = z 2 , by (3.10), we have 1 2 z 2 + 1 2 z∗ 2 + z∗, z + 1 2 v∗ 2 + 1 2 v 2 + v, v ∗ ≤ 0. Hence z ∗ ∈ −Jz. By (3.9), (x, x∗ ) ∈ gra A + gra(−J). Remark 3.2.16 Haraux provides a very simple proof of Theorem 3.2.15 in Hilbert spaces in [51, Theorem 10], but the proof could not be adapted to reflexive Banach spaces (The proof is based on the application of Minty’s Theorem). 29 3.3. An unbounded skew operator on 3.3 2 (N) An unbounded skew operator on 2 (N) In this section, we construct a maximally monotone and skew operator S 2 (N) on such that −S ∗ is not maximally monotone. This answers Svaiter’s question raised in [80]. We also show its domain is a proper subset of the domain of its adjoint S ∗ , i.e., dom S dom S ∗ . Throughout this section, H denotes a Hilbert space. Section 3.3 is all based on the work in [17] by Bauschke, Wang and Yao . Let 2 (N) denote the Hilbert space of real square-summable sequences 2 i≥1 xi (xn )n∈N = (x1 , x2 , x3 , . . .) with Example 3.3.1 Let H = i<n yi Sy = 2 (N), − < +∞. and S : dom S → 2 (N) be given by i>n yi n∈N 2 yi + 12 yn = i<n , n∈N ∀y = (yn )n∈N ∈ dom S, where dom S = y = (yn ) ∈ and i<1 yi 2 (N) (3.11) | i≥1 yi = 0, i≤n yi n∈N ∈ 2 (N) is understood to mean 0. In matrix form, 0 −1 −1 −1 −1 · · · 1 0 −1 −1 −1 · · · 1 1 0 −1 −1 · · · 1 S= 2 1 1 1 0 −1 · · · 1 1 1 1 0 ··· . . . . . .. . . . . . . . . −1 −1 · · · −1 −1 · · · −1 −1 · · · , −1 −1 · · · −1 −1 · · · 30 2 (N) 3.3. An unbounded skew operator on or 1 2 1 1 S= 1 1 . .. 0 0 0 0 ··· 1 2 0 0 0 ··· 1 1 2 0 0 ··· 1 1 1 2 0 ··· 1 1 1 1 2 .. .. .. .. ··· . . . 0 0 · · · 0 0 · · · 0 0 · · · . 0 0 · · · 0 0 · · · . Using the second matrix, it is easy to see that S is injective. Proposition 3.3.2 Let S be defined as in Example 3.3.1. Then S is skew. Proof. Let x = (xn )n∈N ∈ dom S. Then 2 xi (N) n∈N i≤n i≤n xi n∈N − 12 x = xi n∈N i≤n ∈ Thus, − 12 (xn )n∈N xi + 12 xn = 2 (N). i<n = Sx. n∈N Hence S is well defined. Clearly, S is linear on dom S. Now we show S is skew. Let y = (yn )n∈N ∈ dom S, and s = 2 (N). i<n yi Hence = i≥1 yi . n∈N i≤n yi n∈N i≤n yi Then n∈N − (yn )n∈N ∈ 2 (N). ∈ Since s = 0, 2 (N) − yi i<n n∈N yi = i≥n =0− yi i<n = n∈N i≥1 yi − yi i<n n∈N , n∈N 31 3.3. An unbounded skew operator on yi n∈N i≥n+1 =0− yi i≤n n∈N ∈ 2 2 (N) (N). (3.12) Thus, by (3.12), − 2 Sy, y = i>n yi − yi ,y i<n yi + = n∈N i≥n+1 yi ,y n∈N i≥n (3.13) yi , = i≥1 yi , . . . i≥2 yi , + i≥2 yi , . . . , y i≥3 = (s, s − y1 , s − (y1 + y2 ), . . .) + (s − y1 , s − (y1 + y2 ), . . .), (y1 , y2 , . . .) = [sy1 + (s − y1 )y2 + (s − (y1 + y2 ))y3 + · · · ]+ [(s − y1 )y1 + (s − (y1 + y2 ))y2 + (s − (y1 + y2 + y3 ))y3 + · · · ] = lim[sy1 + (s − y1 )y2 + · · · + (s − (y1 + · · · + yn−1 ))yn ]+ n lim[(s − y1 )y1 + (s − (y1 + y2 ))y2 + · · · + (s − (y1 + · · · + yn ))yn ] n = lim[s(y1 + · · · + yn ) − y1 y2 − (y1 + y2 )y3 − · · · − (y1 + · · · + yn−1 )yn ]+ n [s(y1 + · · · + yn ) − (y12 + · · · + yn2 ) − y1 y2 − · · · − (y1 + · · · + yn−1 )yn ] = lim[2s(y1 + · · · + yn ) − (y1 + · · · + yn )2 ] = 2s2 − s2 = s2 = 0. n Hence S is skew. Proposition 3.3.3 Let S be defined as in Example 3.3.1. Then S is a maximally monotone operator. In particular, gra S is closed. Proof. By Proposition 3.3.2, S is skew. Let (x, x∗ ) ∈ 2 (N) × 2 (N) be monotonically related to gra S. Write x = (xn )n∈N and x∗ = (x∗n )n∈N . By 32 2 (N) 3.3. An unbounded skew operator on Fact 3.2.9, we have Sy, x + x∗ , y = 0, ∀y ∈ dom S. (3.14) Let en = (0, . . . , 0, 1, 0, . . .) : the nth entry is 1 and the others are 0. Then let y = −e1 + en . Thus y ∈ dom S and Sy = (− 12 , −1, . . . , −1, − 12 , 0, . . .). Then by (3.14), n−1 − x∗1 + x∗n − 1 2 x1 − 1 2 xn − i=2 n−1 xi = 0 ⇒ x∗n = x∗1 − 1 2 x1 xi + 21 xn . + i=1 (3.15) Since x∗ ∈ 2 (N) 2 (N), and x ∈ − Next we show − i≥1 xi we have x∗n → 0, xn → 0. Thus by (3.15), i≥1 xi = x∗1 − 12 x1 . (3.16) = x∗1 − 12 x1 = 0. Let s = i≥1 xi . Then by (3.15) and (3.16), 2x∗ = 2(x∗n )n∈N = 2 − = = = −2 −2 − i≥1 xi + 2 i≥1 i<n n∈N − = xi + xn xi − n∈N xi + xn i<n n∈N i≥n i≥n xi + 12 xn xi + . xi i≥n+1 i≥n xi − xi + xn i≥n n∈N (3.17) n∈N 33 2 (N) 3.3. An unbounded skew operator on On the other hand, by (3.15) and (3.16), 2 (N) x∗ − 12 x = − xi + 12 xn xi + i≥1 n∈N i<n − ( 12 xn )n∈N = − n∈N Then by (3.17), 2x∗ = − xi i≥n − + n∈N xi . n∈N i≥n+1 Then by Fact 3.2.9, similar to the proof in (3.13) in Proposition 3.3.2, we have 0 ≥ −2 x∗ , x = xi xi , = i≥1 xi + n∈N i≥n i≥n+1 xi , . . . ,x n∈N xi , + i≥2 i≥2 xi , . . . , x i≥3 = 2s2 − s2 = s2 . 1 i<n xi + 2 xn Hence s = 0, i.e., x∗1 = 12 x1 by (3.16). By (3.15), x∗ = . n∈N Thus 2 (N) x∗ + 12 x = xi + 12 xn i<n + n∈N 1 2 xn n∈N xi = i≤n . xi i≥n . n∈N Hence x ∈ dom S and x∗ = Sx. Thus, S is maximally monotone. Hence gra S is closed. Remark 3.3.4 Let S be as in Example 3.3.1. Since e1 = (1, 0, 0, . . . , 0, . . .) ∈ / dom S, the operator S is unbounded. 34 2 (N) 3.3. An unbounded skew operator on Proposition 3.3.5 Let S be defined as in Example 3.3.1. Then S∗y = yi + 12 yn n∈N i>n where dom S ∗ = 2 (N) ∀y = (yn )n∈N ∈ dom S ∗ , , 2 (N) y = (yn )n∈N ∈ | i≥1 yi ∈ R, (3.18) i>n yi n∈N ∈ . In matrix form, 1 2 0 0 ∗ S = 0 0 . .. Moreover, dom S 1 1 1 1 ··· 1 1 1 2 1 1 1 ··· 1 1 0 1 2 1 1 ··· 1 1 0 0 1 2 1 ··· 1 1 0 0 0 1 2 ··· .. .. . . 1 1 ··· ··· .. . i>n yi + . .. . · · · · · · · · · . · · · · · · dom S ∗ , S ∗ = −S on dom S, and S ∗ is not skew. Proof. Let y = (yn )n∈N ∈ 1 2 yn .. 2 (N) i>n yi with n∈N . Now we show (y, y ∗ ) n∈N ∈ 2 (N), ∈ gra S ∗ . Let s = and y ∗ = i≥1 yi and x ∈ dom S. Then we have y, Sx + y ∗ , −x = = xi y, i<n y, 12 x + yi i>n + n∈N i<n + n∈N xi n∈N 1 2y yi + i>n n∈N , −x , −x = lim [y2 x1 + y3 (x1 + x2 ) + · · · + yn (x1 + · · · + xn−1 )] n − lim [x1 (s − y1 ) + x2 (s − y1 − y2 ) + · · · + xn (s − y1 − · · · − yn )] n 35 3.3. An unbounded skew operator on 2 (N) = lim [x1 (y2 + · · · + yn ) + x2 (y3 + · · · + yn ) + · · · + xn−1 yn ] n − lim [x1 (s − y1 ) + x2 (s − y1 − y2 ) + · · · + xn (s − y1 − · · · − yn )] n = lim [x1 (y1 + y2 + · · · + yn − s) + x2 (y1 + y2 + · · · + yn − s) + · · · n + xn (y1 + y2 + · · · + yn − s)] = lim [(x1 + · · · + xn )(y1 + y2 + · · · + yn − s)] n = 0. Hence (y, y ∗ ) ∈ gra S ∗ . On the other hand, let (a, a∗ ) ∈ gra S ∗ with a = (an )n∈N and a∗ = (a∗n )n∈N . Now we show ai i>n n∈N ∈ 2 (N) and a∗ = ai + 12 an i>n . (3.19) n∈N Let en = (0, . . . , 0, 1, 0, . . .) : the nth entry is 1 and the others are 0. Then let y = −e1 + en . Thus y ∈ dom S and Sy = (− 12 , −1, . . . , −1, − 12 , 0, . . .). Then, n−1 0 = a∗ , y + −Sy, a = −a∗1 + a∗n + 12 a1 + 12 an + ai i=2 n−1 ⇒ a∗n = a∗1 − 21 a1 − Since a∗ ∈ 2 (N) and a ∈ i=2 2 (N), ai − 21 an . (3.20) a∗n → 0, an → 0. Thus by (3.20), a∗1 = 12 a1 + ai , (3.21) i>1 36 2 (N) 3.3. An unbounded skew operator on from which we see that i≥1 ai ∈ R. Combining (3.20) and (3.21), we have a∗n = ai + 12 an i>n Thus, (3.19) holds. Hence (3.18) holds. Now for x ∈ dom S, since S∗x = 1 2 xn i≥1 xi + xi = n∈N i>n = = 0, we have − 12 xn − xi i<n n∈N − 12 xn + xi n∈N i≥n = −Sx. We note that S ∗ is not skew since for e1 = (1, 0, . . .), S ∗ e1 , e1 = 1/2e1 , e1 = 1/2. As e1 = (1, 0, 0, . . . , 0, . . .) ∈ dom S ∗ but e1 ∈ dom S, we have dom S dom S ∗ . Proposition 3.3.6 Let S be defined as in Example 3.3.1. Then S ∗ y, y = 12 s2 , ∀y ∈ dom S ∗ with Proof. Let y = (yn )n∈N ∈ dom S ∗ , and s = s= yi . (3.22) i≥1 i≥1 yi . By Proposition 3.3.5, we have s ∈ R and S ∗ y, y = yi + 12 yn i>n ,y = n∈N i≥n yi − 21 yn ,y n∈N = lim[sy1 + (s − y1 )y2 + · · · + (s − y1 − y2 − · · · − yn−1 )yn n − 21 (y12 + y22 + · · · + yn2 )] = lim[s(y1 + · · · + yn ) − y1 y2 − (y1 + y2 )y3 − · · · n 37 3.3. An unbounded skew operator on − (y1 + y2 + · · · + yn−1 )yn ] − 1 2 2 (N) y12 + y22 + · · · + yn2 = lim [s(y1 + · · · + yn )] n − lim[y1 y2 + (y1 + y2 )y3 + · · · + (y1 + y2 + · · · + yn−1 )yn n + 21 (y12 + y22 + · · · + yn2 )] = s2 − lim 12 [y1 + y2 + · · · + yn ]2 n = s2 − 12 s2 = 12 s2 . Hence (3.22) holds. Proposition 3.3.7 Let S be defined as in Example 3.3.1. Then −S is not maximally monotone. Proof. By Proposition 3.3.2, −S is skew. Let e1 = (1, 0, 0, . . . , 0, . . .). Then e1 ∈ / dom S = dom(−S). Thus, (e1 , 12 e1 ) ∈ / gra(−S). We have for every y ∈ dom S, e1 , 12 e1 ≥ 0 and e1 , −Sy + y, 21 e1 = − 12 y1 + 12 y1 = 0. By Fact 3.2.9, (e1 , 21 e1 ) is monotonically related to gra(−S). Hence −S is not maximally monotone. Suppose that X = 2 (N). We proceed to show that for every maximally monotone and skew operator S, the operator −S has a unique maximally monotone extension, namely S ∗ |X . 38 3.3. An unbounded skew operator on 2 (N) Theorem 3.3.8 Let S : dom S → X ∗ be a maximally monotone skew operator. Then −S has a unique maximally monotone extension: S ∗ |X . Proof. By Fact 3.2.12, gra(−S) ⊆ gra S ∗ |X . Assume T is a maximally monotone extension of −S. Let (x, x∗ ) ∈ gra T . Then (x, x∗ ) is monotonically related to gra(−S). By Fact 3.2.9, x∗ , y + −x, Sy = x∗ , y + x, −Sy = 0, ∀y ∈ dom S. Thus (x, x∗ ) ∈ gra S ∗ |X . Since (x, x∗ ) ∈ gra T is arbitrary, we have gra T ⊆ gra S ∗ |X . By Fact 3.2.10(iii), S ∗ |X is monotone. Hence T = S ∗ |X . Remark 3.3.9 Note that [87, Proposition 17] also implies that −S has a unique maximally monotone extension, where S is as in Theorem 3.3.8. Remark 3.3.10 Define the right and left shift operators R, L : 2 (N) 2 (N) → by Rx = (0, x1 , x2 , . . .), Lx = (x2 , x3 , . . .), ∀ x = (x1 , x2 , . . .) ∈ 2 (N). One can verify that in Example 3.3.1 S = (Id −R)−1 − Id , 2 S ∗ = (Id −L)−1 − Id . 2 The maximally monotone operators (Id −R)−1 and (Id −L)−1 have been utilized by Phelps and Simons, see [63, Example 7.4]. 39 3.3. An unbounded skew operator on 2 (N) Example 3.3.11 (S + S ∗ fails to be maximally monotone) Let S be defined as in Example 3.3.1. Then neither S nor S ∗ has full domain. By Fact 3.2.12, ∀x ∈ dom(S + S ∗ ) = dom S, we have (S + S ∗ )x = 0. Thus S +S ∗ has a proper monotone extension from dom(S +S ∗ ) to the 0 map on 2 (N). Consequently, S + S ∗ is not maximally monotone. This supplies a different example for showing that the constraint qualification in the sum problem of maximal monotone operators cannot be substantially weakened, see [63, Example 7.4]. Svaiter introduced S in [80], which is defined by gra S = (x, x∗ ) ∈ X × X ∗ | (x∗ , x) ∈ (gra S)⊥ . Hence S = −S ∗ |X . Definition 3.3.12 Let S : X ⇒ X ∗ be skew. We say S is maximally skew (termed “maximal self-cancelling” in [80]) if no proper enlargement (in the sense of graph inclusion) of S is skew. We say T is a maximally skew extension of S if T is maximally skew and gra T ⊇ gra S. Lemma 3.3.13 Let S : X ⇒ X ∗ be a maximally monotone skew operator. Then both S and −S are maximally skew. Proof. Clearly, S is maximally skew. Now we show −S is maximally skew. Let T be a skew operator such that gra(−S) ⊆ gra T . Thus, gra S ⊆ 40 3.3. An unbounded skew operator on 2 (N) gra(−T ). Since −T is monotone and S is maximally monotone, gra S = gra(−T ). Then −S = T . Hence −S is maximally skew. Fact 3.3.14 (Svaiter) (See [80].) Let S : X ⇒ X ∗ be maximally skew. Then either −S ∗ |X (i.e., S ) or S ∗ |X (i.e., − S ) is maximally monotone. In [80], Svaiter asked whether or not −S ∗ |X (i.e., S ) is maximally monotone if S is maximally skew. Now we can give a negative answer, even though S is maximally monotone and skew. Theorem 3.3.15 Let S be defined as in Example 3.3.1. Then S is maximally skew, but −S ∗ is not monotone, so not maximally monotone. Proof. Let e1 = (1, 0, 0, . . . , 0, . . .). By Proposition 3.3.5, (e1 , − 12 e1 ) ∈ gra(−S ∗ ), but e1 , − 12 e1 = − 12 < 0. Hence −S ∗ is not monotone. By Theorem 3.3.15, −S ∗ |X (i.e., S ) is not always maximally monotone. Can one improve Svaiter’s result to: “If S is maximally skew, then S ∗ |X (i.e., −S ) is always maximally monotone”? Theorem 3.3.16 There exists a maximally skew operator T on 2 (N) such that T ∗ is not maximally monotone. Consequently, Svaiter’s result is optimal. Proof. Let T = −S, where S be defined as in Example 3.3.1. By Lemma 3.3.13, T is maximally skew. Then by Theorem 3.3.15 and Proposition 3.1.3(x), T ∗ = (−S)∗ = −S ∗ is not maximally monotone. Hence Svaiter’s result cannot be further improved. 41 3.4. The inverse Volterra operator on L2 [0, 1] 3.4 The inverse Volterra operator on L2 [0, 1] Section 3.4 is all based on the work in [17] by Bauschke, Wang and Yao . Let V be the Volterra integral operator. In this section, we systematically study T = V −1 and its skew part S = 21 (T − T ∗ ). It turns out that T is neither skew nor symmetric and that its skew part S admits two maximally monotone and skew extensions T1 , T2 (in fact, anti-self-adjoint) even though dom S is a dense linear subspace of L2 [0, 1]. This will give another simpler example of Phelps-Simons’ showing that the constraint qualification for the sum of monotone operators cannot be significantly weakened, see [78, Theorem 5.5] or [83]. Definition 3.4.1 ([15]) Let A : H ⇒ H be a linear relation. We say that A is anti-self-adjoint if A∗ = −A. To study the Volterra operator and its inverse, we shall frequently need the following generalized integration-by-parts formula, see [79, Theorem 6.90]. Fact 3.4.2 (Generalized integration by parts) Assume that x, y are absolutely continuous functions on the interval [a, b]. Then b b xy + a a x y = x(b)y(b) − x(a)y(a). Fact 3.2.13 allows us to claim the following proposition. Proposition 3.4.3 Let A : H ⇒ H be a linear relation. If A∗ = −A, then both A and −A are maximally monotone and skew. 42 3.4. The inverse Volterra operator on L2 [0, 1] Proof. Since A = −A∗ , we have that dom A = dom A∗ and that A has closed graph. Now ∀x ∈ dom A, by Proposition 3.1.3(v), Ax, x = x, A∗ x = − x, Ax ⇒ Ax, x = 0. Hence A and −A are skew. As A∗ = −A is monotone, Fact 3.2.13 shows that A is maximally monotone. Now −A = A∗ = −(−A)∗ and −A is a linear relation. Similar arguments show that −A is maximally monotone. Example 3.4.4 (Volterra operator) (See [5, Example 3.3].) Set H = L2 [0, 1]. The Volterra integration operator [52, Problem 148] is defined by t V : H → H : x → V x, where V x : [0, 1] → R : t → x, (3.23) 0 and its adjoint is given by t → (V ∗ x)(t) = 1 x, t ∀x ∈ X. Then (i) Both V and V ∗ are maximally monotone since they are monotone, continuous and linear. (ii) Both ranges ran V = {x ∈ L2 [0, 1] : x is absolutely continuous, x(0) = 0, x ∈ L2 [0, 1]}, (3.24) 43 3.4. The inverse Volterra operator on L2 [0, 1] and ran V ∗ = {x ∈ L2 [0, 1] : x is absolutely continuous, x(1) = 0, x ∈ L2 [0, 1]}, (3.25) are dense in L2 [0, 1], and both V and V ∗ are one-to-one. (iii) ran V ∩ ran V ∗ = {V x | x ∈ e⊥ }, where e ≡ 1 ∈ L2 [0, 1]. (iv) Define V+ x = 21 (V + V ∗ )(x) = 1 2 e, x e. Then V+ is self-adjoint and ran V+ = span{e}. (v) Define V◦ x = 1 2 (V − V ∗ )(x) : t → 1 t 2[ 0 x − 1 t x] ∀x ∈ L2 [0, 1], t ∈ [0, 1]. Then V◦ is anti-self-adjoint and ran V◦ = {x ∈ L2 [0, 1] : x is absolutely continuous on [0, 1], x ∈ L2 [0, 1], x(0) = −x(1)}. Proof. (i) By Fact 3.4.2, 1 t x(s)dsdt = x(t) x, V x = 0 0 1 2 2 1 x(s)ds 0 ≥ 0, so V is monotone. As dom V = L2 [0, 1] and V is continuous, dom V ∗ = L2 [0, 1]. Let x, y ∈ 44 3.4. The inverse Volterra operator on L2 [0, 1] L2 [0, 1]. We have 1 x(t)dt x(s)dsy(t)dt = 0 0 0 1 1 = 0 0 1 1 1 t V x, y = 0 y(s)ds − 1 t y(s)ds − y(s)ds x(t)dt = y(s)dsx(t)dt 0 1 0 y(s)dsx(t)dt 0 0 t t = V ∗ y, x , thus (V ∗ y)(t) = 1 t y(s)ds ∀t ∈ [0, 1]. (ii) To show (3.24), if z ∈ ran V , then t x z(t) = 0 for some x ∈ L2 [0, 1], and hence z(0) = 0, z is absolutely continuous, and z = x ∈ L2 [0, 1]. On the other hand, if z(0) = 0, z is absolutely continuous, z ∈ L2 [0, 1], then z =Vz . To show (3.25), if z ∈ ran V ∗ , then 1 x z(t) = t for some x ∈ L2 [0, 1], and hence z(1) = 0, z is a absolutely continuous, and z = −x ∈ L2 [0, 1]. On the other hand, if z(1) = 0, z is absolutely continuous, z ∈ L2 [0, 1], then z = V ∗ (−z ). (iii) follows from (ii) (or see [5]). (iv) is clear. (v) If x is absolutely continuous, x(0) = −x(1), x ∈ L2 [0, 1], we have 45 3.4. The inverse Volterra operator on L2 [0, 1] V◦ x (t) = 1 2 1 t 0 x − x = t 1 2 x(t) − x(0) − x(1) + x(t) = x(t). This shows that x ∈ ran V◦ . Conversely, if x ∈ ran V◦ , i.e., x(t) = 1 2 t 0 y− 1 2 1 y t for some y ∈ L2 [0, 1], then x is absolutely continuous, x = y ∈ L2 [0, 1] and x(0) = −x(1) = − 21 1 0 y. Theorem 3.4.5 (Inverse Volterra operator) Let H = L2 [0, 1], and V be the Volterra integration operator. We let T = V −1 and D = dom T ∩ dom T ∗ . Then the following hold. (i) T : dom T → X is given by T x = x with dom T = {x ∈ L2 [0, 1] : x is absolutely continuous, x(0) = 0, x ∈ L2 [0, 1]}, and T ∗ : dom T ∗ → L2 [0, 1] is given by T ∗ x = −x with dom T ∗ = {x ∈ L2 [0, 1] : x is absolutely continuous, x(1) = 0, x ∈ L2 [0, 1]}. Both T and T ∗ are maximally monotone linear operators. (ii) T is neither skew nor symmetric. 46 3.4. The inverse Volterra operator on L2 [0, 1] (iii) The linear subspace D = x ∈ L2 [0, 1] : x is absolutely continuous, x(0) = x(1) = 0, x ∈ L2 [0, 1] is dense in L2 [0, 1]. Moreover, T and T ∗ are skew on D. Proof. (i): T and T ∗ are maximally monotone because T = V −1 , and T ∗ = (V −1 )∗ = (V ∗ )−1 and Example 3.4.4(i). By Example 3.4.4(ii), T : L2 [0, 1] → L2 [0, 1] has dom T = {x ∈ L2 [0, 1] : x is absolutely continuous, x(0) = 0, x ∈ L2 [0, 1]} dom T ∗ = {x ∈ L2 [0, 1] : x is absolutely continuous, x(1) = 0, x ∈ L2 [0, 1]} T x = x , ∀x ∈ dom T, T ∗ y = −y and ∀y ∈ dom T ∗ . Note that by Fact 3.4.2, 1 xx= T x, x = 0 1 T ∗ x, x = 0 1 2 1 1 x (1) − x2 (0) = x(1)2 2 2 2 ∀x ∈ dom T, 1 1 1 −x x = −( x(1)2 − x(0)2 ) = x(0)2 2 2 2 (3.26) ∀x ∈ dom T ∗ . (3.27) 47 3.4. The inverse Volterra operator on L2 [0, 1] (ii): Letting x(t) = t, y(t) = t2 we have 1 T x, x = 0 t = 12 , 1 x, T y = 2t2 = 0 2 3 = 1 3 1 = t2 = T x, y 0 ⇒ T x, x = 0, T x, y = x, T y . (iii): By (i), D = dom T ∩dom T ∗ is clearly a linear subspace. For x ∈ D, x(0) = x(1) = 0, from (3.26) and (3.27), T x, x = 21 x(1)2 = 0, T ∗ x, x = 12 x(0)2 = 0. Hence both T and T ∗ are skew on D. The fact that D is dense in L2 [0, 1] follows from [79, Theorem 6.111]. Our proof of (ii), (iii) in the following theorem follows the ideas of [69, Example 13.4]. Theorem 3.4.6 (The skew part of the inverse Volterra operator) Let H = L2 [0, 1], and T be defined as in Theorem 3.4.5. Let S = (i) Sx = x (∀x ∈ dom S) and gra S = {(V x, x) | T −T ∗ 2 . x ∈ e⊥ }, where e ≡ 1 ∈ L2 [0, 1]. In particular, dom S = {x ∈ L2 [0, 1] : x is absolutely continuous, x(0) = x(1) = 0, x ∈ L2 [0, 1]}, ran S = {y ∈ L2 [0, 1] : e, y = 0} = e⊥ . 48 3.4. The inverse Volterra operator on L2 [0, 1] Moreover, dom S is dense, and S −1 = V |e⊥ , (−S)−1 = V ∗ |e⊥ , (3.28) consequently, S is skew, and neither S nor −S is maximally monotone. (ii) The adjoint of S has gra S ∗ = {(V ∗ x∗ + le, x∗ ) | x∗ ∈ L2 [0, 1], l ∈ R}. More precisely, S ∗ x = −x ∀x ∈ dom S ∗ , with dom S ∗ = {x ∈ L2 [0, 1] : x is absolutely continuous on [0, 1], x ∈ L2 [0, 1]}, ran S ∗ = L2 [0, 1]. Neither S ∗ nor −S ∗ is monotone. Moreover, S ∗∗ = S. (iii) Let T1 : dom T1 → L2 [0, 1] be defined by T1 x = x , ∀x ∈ dom T1 , with dom T1 = {x ∈ L2 [0, 1] : x is absolutely continuous, x(0) = x(1), x ∈ L2 [0, 1]}. Then T1∗ = −T1 , ran T1 = e⊥ . (3.29) Hence T1 is skew, and a maximally monotone extension of S; and −T1 is skew and a maximally monotone extension of −S. 49 3.4. The inverse Volterra operator on L2 [0, 1] Proof. (i): By Theorem 3.4.5(iii), we get dom S directly. Now (∀x ∈ dom S = dom T ∩ dom T ∗ ) T x = x and T ∗ x = −x , so Sx = x . Then Example 3.4.4(iii) implies gra S = {(V x, x) | x ∈ e⊥ }. Hence gra S −1 = {(x, V x) | x ∈ e⊥ }. (3.30) Theorem 3.4.5(iii) implies dom S is dense. Furthermore, gra(−S) = {(V x, −x) | x ∈ e⊥ }, so gra(−S)−1 = {(x, −V x) | x ∈ e⊥ }. (3.31) Since t 1 1 1 V ∗ x(t) = x−0 = t x− 0 t x=− 0 x = −V x(t) ∀t ∈ [0, 1] , ∀x ∈ e⊥ we have −V x = V ∗ x, ∀x ∈ e⊥ . Then by (3.31), gra(−S)−1 = {(x, V ∗ x) | x ∈ e⊥ }. (3.32) Hence, (3.30) and (3.32) together establish (3.28). As both V, V ∗ are maximally monotone with full domain, we conclude that S −1 , (−S)−1 are not maximally monotone, thus S, −S are not maximally monotone. (ii): By (i), we have (x, x∗ ) ∈ gra S ∗ ⇔ −x, y + x∗ , V y = 0, ⇔ −x + V ∗ x∗ , y = 0, ∀y ∈ e⊥ ∀y ∈ e⊥ ⇔ x − V ∗ x∗ ∈ span{e}. 50 3.4. The inverse Volterra operator on L2 [0, 1] Equivalently, x = V ∗ x∗ +ke for some k ∈ R. This means that x is absolutely continuous, x∗ = −x ∈ L2 [0, 1] On the other hand, if x is absolutely continuous and x ∈ L2 [0, 1], observe that 1 x(t) = t −x + x(1)e, so that x − V ∗ (−x ) ∈ span{e} and (x, −x ) ∈ gra S ∗ . It follows that dom S ∗ = {x ∈ L2 [0, 1] : x is absolutely continuous on [0, 1], x ∈ L2 [0, 1]}, ran S ∗ = L2 [0, 1], and S ∗ x = −x , ∀x ∈ dom S ∗ . Since S ∗ x, x = − 1 0 xx=− 1 1 x(1)2 − x(0)2 , 2 2 we conclude that neither S ∗ nor −S ∗ is monotone. Now we show S ∗∗ = S. V has closed graph ⇒ V |e⊥ has closed graph ⇒ S −1 has closed graph ⇒ S has closed graph ⇒ gra S = gra S ∗∗ ⇒ S ∗∗ = S. (iii): To show (3.29), suppose that x is absolutely continuous and that x(0) = x(1). Then 1 0 x = x(1) − x(0) = 0 ⇒ T1 x = x ∈ e⊥ . Conversely, if x ∈ L2 [0, 1] satisfies e, x = 0, we define t → z(t) = t 0 x, then z is absolutely continuous, z(0) = z(1), T1 z = x. Hence ran T1 = e⊥ . 51 3.4. The inverse Volterra operator on L2 [0, 1] T1 is skew, because for every x ∈ dom T1 , we have 1 T1 x, x = 0 x x = 21 x(1)2 − 21 x(0)2 = 0. Moreover, T1∗ = −T1 : indeed, as T1 is skew, by Fact 3.2.12, gra(−T1 ) ⊆ gra T1∗ . To show that T1∗ = −T1 , take z ∈ dom T1∗ , ϕ = T1∗ z. Put Φ(t) = t 0 ϕ. We have ∀y ∈ dom T1 , 1 0 y z = T1 y, z = T1∗ z, y = ϕ, y = 1 1 yΦ yϕ = 0 (3.33) 0 1 = [Φ(1)y(1) − Φ(0)y(0)] − Φy . (3.34) 0 Using y = e ∈ dom T1 gives Φ(1) − Φ(0) = 0, from which Φ(1) = Φ(0) = 0. It follows from (3.33)–(3.34) that 1 0 y (z + Φ) = 0 ∀y ∈ dom T1 . Since ran T1 = e⊥ , z + Φ ∈ span{e}, say z + Φ = ke for some constant k ∈ R. Then z is absolutely continuous, z(0) = z(1) since Φ(0) = Φ(1) = 0, and T1∗ z = ϕ = Φ = −z . This implies that dom T1∗ ⊆ dom T1 . Then by Fact 3.2.12, T1∗ = −T1 . It remains to apply Proposition 3.4.3. Remark 3.4.7 Let S be defined in Theorem 3.4.6. Now we give a new proof to show that S ∗∗ = S in Theorem 3.4.6 (ii). Applying similar arguments as [42, Example 8.22], one can indeed show that S has a closed graph, so S ∗∗ = S. Or, by [63, Proposition 3.2(e)], S has a closed graph, then S ∗∗ = S. Fact 3.4.8 Let H be a Hilbert space and A : H ⇒ H. Then (−A)−1 = 52 3.4. The inverse Volterra operator on L2 [0, 1] A−1 ◦ (− Id). If A is a linear relation, then (−A)−1 = −A−1 . Proof. This follows from the definition of the set-valued inverse. Indeed, x ∈ (−A)−1 (x∗ ) ⇔ (x, x∗ ) ∈ gra(−A) ⇔ (x, −x∗ ) ∈ gra A ⇔ x ∈ A−1 (−x∗ ). When A is a linear relation, x ∈ (−A)−1 (x∗ ) ⇔ (x, −x∗ ) ∈ gra A ⇔ (−x, x∗ ) ∈ gra A ⇔ −x ∈ A−1 x∗ ⇔ x ∈ −A−1 (x∗ ). Theorem 3.4.9 (The inverse of the skew part of Volterra operator) Let H = L2 [0, 1], and V be the Volterra integration operator, and V◦ : L2 [0, 1] → L2 [0, 1] be given by V◦ = V −V∗ . 2 Define T2 : dom T2 → L2 [0, 1] by T2 = V◦−1 . Then (i) T2 x = x , ∀x ∈ dom T2 where dom T2 = {x ∈ H : x is absolutely continuous on [0, 1], x ∈ H, x(0) = −x(1)}. (3.35) (ii) T2∗ = −T2 , and both T2 , −T2 are maximally monotone and skew. Proof. (i): Since V◦ x(t) = 1 2 1 t 0 x− x , t 53 3.4. The inverse Volterra operator on L2 [0, 1] V◦ is a one-to-one map. Then V◦−1 1 ( 2 1 t 0 x− x) = x(t) = t 1 ( 2 1 t 0 x− x) , t which implies T2 x = V◦−1 x = x for x ∈ ran V◦ . As dom T2 = ran V◦ , by Example 3.4.4(v), ran V◦ can be written as (3.35). (ii): Since dom V = dom V ∗ = L2 [0, 1], V◦ is skew on L2 [0, 1], so maximally monotone. Then T2 = V◦−1 is maximally monotone. Since V◦ is skew and dom V◦ = L2 [0, 1], we have V◦∗ = −V◦ , by Fact 3.4.8, T2∗ = (V◦−1 )∗ = (V◦∗ )−1 = (−V◦ )−1 = −V◦−1 = −T2 . By Proposition 3.4.3, we have both T2 and −T2 are maximally monotone and skew. Remark 3.4.10 Note that while V◦ is continuous on L2 [0, 1], the operator S given in Theorem 3.4.6 is discontinuous. Combining Theorem 3.4.5, Theorem 3.4.6 and Theorem 3.4.9, we can summarize the relationships among the differentiation operators encountered in this section. Corollary 3.4.11 Let T be defined in Theorem 3.4.5 and S, T1 be defined in Theorem 3.4.6 and T2 be defined in Theorem 3.4.9. Then the domain of the skew operator S is dense in L2 [0, 1]. Neither S nor −S is maximally monotone. Neither S ∗ nor −S ∗ is monotone. 54 3.4. The inverse Volterra operator on L2 [0, 1] The linear operators S, T, T1 , T2 satisfy: gra S gra T gra(−S ∗ ), gra S gra T1 gra(−S ∗ ), gra S gra T2 gra(−S ∗ ). While S is skew, T, T1 , T2 are maximally monotone and T1 , T2 are skew. Also, gra(−S) gra(T ∗ ) gra S ∗ , gra(−S) gra(−T1 ) gra S ∗ , gra(−S) gra(−T2 ) gra S ∗ . While −S is skew, T ∗ , −T1 , −T2 are maximally monotone and −T1 , −T2 are skew. Remark 3.4.12 (i): Note that while T1 , T2 are maximally monotone, −T1 , −T2 are also maximally monotone. This is in stark contrast with the maximally monotone skew operator given in Proposition 3.3.3 and Proposition 3.3.7 such that its negative is not maximally monotone. (ii): Even though the skew operator S in Theorem 3.4.6 has dom S dense in L2 [0, 1], it still admits two distinct maximally monotone and skew extensions T1 , T2 . Example 3.4.13 (T + T ∗ fails to be maximally monotone) Let T be defined as in Theorem 3.4.5, and T1 , T2 be respectively defined in Theo55 3.5. Discussion rem 3.4.6 and Theorem 3.4.9. Now ∀x ∈ dom T ∩ dom T ∗ , we have T x + T ∗ x = x − x = 0. Thus T + T ∗ has a proper monotone extension from dom T ∩ dom T ∗ L2 [0, 1] to the 0 map on L2 [0, 1]. Consequently, T + T ∗ is not maximally monotone. Note that dom T ∩ dom T ∗ is dense in L2 [0, 1] and that dom T − dom T ∗ is a dense subspace of L2 [0, 1]. This supplies a simpler example for showing that the constraint qualification in the sum problem of maximally monotone operators cannot be substantially weakened, see [63, Example 7.4]. Similarly, by Theorems 3.4.6 and Theorem 3.4.9, Ti∗ = −Ti , we conclude that Ti + Ti∗ = 0 on dom Ti , a dense subset of L2 [0, 1]; thus, Ti + Ti∗ fails to be maximally monotone while both Ti , Ti∗ are maximally monotone. 3.5 Discussion The Brezis-Browder Theorem (see Fact 3.2.13) is a very important characterization of maximal monotonicities of monotone relations. The original proof [30] is based on the application of Zorn’s Lemma by constructing a series of finite-dimensional subspaces, which is complicated. In Theorem 3.2.15, we establish the Brezis-Browder Theorem by considering the fact that a lower semicontinuous, convex and coercive function on a reflexive space has at least one minimizer. In [75], Simons generalized the Brezis-Browder Theorem to SSDB spaces. The Brezis-Browder Theorem and Corollary 3.2.6 are essential tools for the construction of maximally monotone linear subspace 56 3.5. Discussion extensions of a monotone linear relation, which will be discussed in detail in Chapter 4. There will be an interesting question for the future work on the BrezisBrowder Theorem in a general Banach space: Let A : X ⇒ X ∗ be a monotone linear relation such that gra A is closed. Assume A∗ |X is monotone. Is A necessarily maximally monotone? In Sections 3.3 and 3.4, some explicit monotone linear relations were constructed in Hilbert spaces, which gave a negative answer to a question raised by Svaiter [80] and which showed that the constraint qualification in the sum problem for maximally monotone operators cannot be weakened (see [63, Example 7.4]). In particular, these two sections will provide concrete examples for the characterization of decomposable monotone linear relations discussed in Chapter 9. 57 Chapter 4 Maximally monotone extensions of monotone linear relations This chapter is based on [88] by Wang and Yao. We consider the linear relation G : Rn ⇒ Rn : gra G = {(x, x∗ ) ∈ Rn × Rn | Ax + Bx∗ = 0} where (4.1) A, B ∈ Rp×n , (4.2) rank(A B) = p. (4.3) Our main concern is to find explicit extensions of G that are maximally monotone linear relations. Recently, finding constructive maximally monotone extensions, instead of using Zorn’s lemma, has been a very active topic [11, 13, 39–41]. In [39], Crouzeix and Oca˜ na-Anaya gave an algorithm for finding maximally monotone linear subspace extensions of G, but it is not clear what the maximally monotone extensions are analytically. In this chapter, we provide some maximally monotone extensions of G with closed 58 4.1. Auxiliary results on linear relations analytical forms. Along the way, we also give a new proof of Crouzeix and Oca˜ na-Anaya’s characterizations on monotonicity and maximal monotonicity of G. Our key tool is the Brezis-Browder characterization of maximally monotone linear relations. In this chapter, we use the following notation. Counting multiplicities, let λ1 , λ2 , . . . , λk be all positive eigenvalues of (AB + BA ) and (4.4) λk+1 , λk+2 , . . . , λp be nonpositive eigenvalues of (AB + BA ). (4.5) Moreover, let vi be an eigenvector of eigenvalue λi of (AB +BA ) satisfying vi = 1, and vi , vj = 0 for 1 ≤ i = j ≤ q. It will be convenient to put λ1 0 Idλ = diag(λ1 , . . . , λp ) = 0 . . . 0 4.1 0 0 ··· λ2 0 ··· 0 λ3 0 0 .. 0 0 0 . 0 0 .. . , 0 λp V = [v1 v2 . . . vp ] . (4.6) Auxiliary results on linear relations In this section, we collect some facts and preliminary results which will be used in the sequel. We first provide a result about subspaces on which a linear operator from Rn to Rn , i.e, an n × n matrix, is monotone. For M ∈ Rn×n , define 59 4.1. Auxiliary results on linear relations three subspaces of Rn , namely, the positive eigenspace, null eigenspace and negative eigenspace associated with M + M by V+ (M ) = span V0 (M ) = span V− (M ) = span w1 , . . . , ws : wi is an eigenvector associated with a positive eigenvalue αi of M + M wi , wj = 0 ∀ i = j, wi = 1, i, j = 1, . . . , s. ws+1 , . . . , wl : wi is an eigenvector associated with the 0 eigenvalue of M + M wl+1 , . . . , wn : wi , wj = 0 ∀ i = j, wi = 1, i, j = s + 1, . . . , l. wi is an eigenvector associated with a negative eigenvalue αi of M + M wi , wj = 0 ∀ i = j, wi = 1, i, j = l + 1, . . . , n. which is possible since a symmetric matrix always has a complete orthonormal set of eigenvectors, [59, pages 547–549]. Proposition 4.1.1 Let M be an n × n matrix. Then (i) M is strictly monotone on V+ (M ). Moreover, M + M : V+ (M ) → V+ (M ) is a bijection. (ii) M is monotone on V+ (M ) + V0 (M ). (iii) −M is strictly monotone on V− (M ). Moreover, −(M +M ) : V− (M ) → 60 4.1. Auxiliary results on linear relations V− (M ) is a bijection. (iv) −M is monotone on V− (M ) + V0 (M ). (v) For every x ∈ V0 (M ), (M + M )x = 0 and x, M x = 0. In particular, the orthogonal decomposition holds: Rn = V+ (M ) ⊕ V0 (M ) ⊕ V− (M ). s i=1 li wi Proof. (i): Let x ∈ V+ (M ). Then x = for some (l1 , . . . , ls ) ∈ Rs . Since {w1 , · · · , ws } is a set of orthonormal vectors, they are linearly independent so that ⇔ x=0 (l1 , . . . , ls ) = 0. Note that αi > 0 when i = 1, . . . , s and wi , wj = 0 for i = j. We have s s i=1 i=1 i=1 i=1 αi li2 > 0 li αi wi = li wi , = s s s li wi ) li wi , (M + M )( 2 x, M x = x, (M + M )x = i=1 if x = 0. For every x ∈ V+ (M ) with x = s i=1 li wi , we have s s li (M + M )wi = (M + M )x = i=1 i=1 αi li wi ∈ V+ (M ). As αi > 0 for i = 1, . . . , s and {w1 , . . . , ws } is an orthonormal basis of V+ (M ), we conclude that M + M : V+ (M ) → V+ (M ) is a bijection. 61 4.1. Auxiliary results on linear relations (ii): Let x ∈ V+ (M )+V0 (M ). Then x = l i=1 li wi for some (l1 , . . . , ll ) ∈ Rl . Note that αi ≥ 0 when i = 1, . . . , l and wi , wj = 0 for i = j. We have l l i=1 i=1 l l l li αi wi = li wi , = i=1 li wi ) li wi , (M + M )( 2 x, M x = x, (M + M )x = i=1 i=1 αi li2 ≥ 0. The proofs for (iii), (iv) are similar to (i), (ii). (v): Obvious. Corollary 4.1.2 The following hold: (i) gra T = {(B u, A u) | u ∈ V+ (BA )} is strictly monotone. (ii) gra T = {(B u, A u) | u ∈ V+ (BA ) + V0 (BA )} is monotone. (iii) gra T = {(B u, −A u) | u ∈ V− (BA )} is strictly monotone. (iv) gra T = {(B u, −A u) | u ∈ V− (BA ) + V0 (BA ))} is monotone. 62 4.1. Auxiliary results on linear relations Proof. As B u, A u = u, BA u ∀u ∈ Rn , the result follows from Proposition 4.1.1 by letting M = BA . Lemma 4.1.3 For every subspace S ⊆ Rp , the following hold. dim{(B u, A u) | u ∈ S} = dim S. (4.7) dim{(B u, −A u) | u ∈ S} = dim S. (4.8) Proof. See [59, page 208, Exercise 4.4.9]. The following fact is straightforward from the definition of V . Fact 4.1.4 We have (AB + BA )V = V Idλ . Some basic properties of G are: Lemma 4.1.5 (i) gra G = ker(A B). (ii) G0 = ker B, G−1 (0) = ker A. (iii) dom G = PX (ker(A B)) and ran G = PX ∗ (ker(A B)). (iv) ran(G + Id) = PX ∗ (ker(A − B B)) = PX (ker(A B − A)), and dom G = PX (ker(A − B B)), ran G = PX ∗ (ker(A (B − A)). 63 4.1. Auxiliary results on linear relations (v) dim gra G = 2n − p. Proof. (i), (ii), (iii) follow from the definition of G. Since Ax+Bx∗ = 0 ⇔ (A−B)x+B(x+x∗ ) = 0 ⇔ A(x+x∗ )+(B−A)x∗ = 0, (iv) holds. (v): We have A 2n = dim ker(A B) + dim ran = dim gra G + p. B Hence dim gra G = 2n − p. The following result summarizes the monotonicities of G∗ and G. Lemma 4.1.6 The following hold. (i) gra G∗ = {(B u, −A u) | u ∈ Rp }. (ii) G∗ is monotone ⇔ the matrix A B+B A ∈ Rp×p is negative-semidefinite. (iii) Assume G is monotone. Then n ≤ p. Moreover, G is maximally monotone if and only if dim gra G = n = p. Proof. (i): By Lemma 4.1.5(i), we have A (x, x∗ ) ∈ gra G∗ ⇔ (x∗ , −x) ∈ gra G⊥ = ran = {(A u, B u) | u ∈ Rp }. B Thus gra G∗ = {(B u, −A u) | u ∈ Rp }. 64 4.1. Auxiliary results on linear relations (ii): Since gra G∗ is a linear subspace, by (i), G∗ is monotone ⇔ B u, −A u ≥ 0, ⇔ u, −BA u ≥ 0, ⇔ u, BA u ≤ 0, ∀u ∈ Rp ∀u ∈ Rp ∀u ∈ Rp ⇔ u, (A B + B A)u ≤ 0, ∀u ∈ Rp ⇔ (A B + B A) is negative semidefinite. (iii): By Fact 3.2.6 and Lemma 4.1.5(v), 2n − p = dim gra G ≤ n ⇒ n ≤ p. By Fact 3.2.6 and Lemma 4.1.5(v) again, G is maximally monotone ⇔ 2n − p = dim gra G = n ⇔ dim gra G = p = n. 4.1.1 One linear relation: two equivalent formulations The linear relation G given by (4.1)–(4.3): gra G = {(x, x∗ ) ∈ Rn × Rn | Ax + Bx∗ = 0} (4.9) is an intersection of p linear hyperplanes. It can be equivalently described as a span of q = 2n−p points in Rn ×Rn . Indeed, for (4.9) we can use Gaussian elimination to reduce (A B) to row echelon form. Then back substitute to solve for the basic variables in terms of the free variables, see [59, page 61]. The row-echelon form gives x = h1 y1 + · · · + h2n−p y2n−p x∗ C = y D 65 4.2. Explicit maximally monotone extensions of monotone linear relations where y ∈ R2n−p and C = (h1 , . . . , h2n−p ) D with C, D being n × (2n − p) matrices. Therefore, Cy C gra G = y ∈ R2n−p = ran Dy D (4.10) which is a span of 2n − p points in Rn × Rn . The two formulations (4.9) and (4.10) coincide when p = q = n, Id = −B = C and D = A in which Id ∈ Rn×n . 4.2 Explicit maximally monotone extensions of monotone linear relations In this section, we give explicit maximally monotone linear subspace extensions of G by using V+ (AB ) or Vg . A characterization of all maximally monotone extensions of G is also given. We also provide a new proof for Crouzeix and Oca˜ na-Anaya’s characterizations of the monotonicity and the maximal monotonicity of G. We shall use notations given in (4.1)–(4.6), in particular, G is in the form of (4.9). Lemma 4.2.1 Let M ∈ Rp×p, and linear relations G and G be defined by gra G = {(x, x∗ ) | M Ax + M Bx∗ = 0} gra G = {(B u, −A u) | u ∈ ran M }. 66 4.2. Explicit maximally monotone extensions of monotone linear relations Then (G)∗ = G. Proof. Let (y, y ∗ ) ∈ Rn × Rn . Then we have (y, y ∗ ) ∈ gra(G)∗ ⇔ (y ∗ , −y) ∈ (gra G)⊥ = (ker M A M B ⇔ (y, y ∗ ) ∈ gra G. ⊥ A M = ran B M Hence (G)∗ = G. Lemma 4.2.2 Define linear relations G and G by gra G = {(x, x∗ ) | Vg Ax + Vg Bx∗ = 0} gra G = {(B u, −A u) | u ∈ V− (BA ) + V0 (BA )}, where Vg is (p − k) × p matrix defined by v k+1 v k+2 Vg = . . . . vp Then (i) G is monotone. (ii) (G)∗ = G. 67 4.2. Explicit maximally monotone extensions of monotone linear relations B (iii) gra G = gra G + u u ∈ V+ (BA A Proof. (i): Apply Corollary 4.1.2(iv). ) . (ii): Notations are as in (4.6). Define the p × p matrix N by 0 0 N = 0 Id in which Id ∈ R(p−k)×(p−k). Then we have N V = 0 0 (v1 · · · vk Vg ) 0 Id 0 = . Vg (4.11) Then we have Vg Ax + Vg Bx∗ = 0 ⇔ 0 Vg Ax + Vg Bx∗ =0 ⇔ N V Ax + N V Bx∗ = 0, ∀(x, x∗ ) ∈ Rn × Rn . Hence gra G = {(x, x∗ ) | N V Ax + N V Bx∗ = 0}. Thus by Lemma 4.2.1 with M = V N , gra(G)∗ = {(B u, −A u) | u ∈ ran V N = ran 0 Vg 68 4.2. Explicit maximally monotone extensions of monotone linear relations = V− (BA ) + V0 (BA )} = gra G. Hence (G)∗ = (G)∗∗ = G. (iii): Let J be defined by B gra J = gra G + u u ∈ V+ (BA A Then we have ) . B ⊥ ⊥ (gra J) = (gra G) ∩ u u ∈ V+ (BA A By Lemma 4.1.5(i), Then if and only if A ⊥ p gra G = w w ∈ R B B A w ∈ u u ∈ V+ (BA A B ⊥ . ) ⊥ ) (A w, B w), (B u, A u) = 0 ∀ u ∈ V+ (BA ), 69 4.2. Explicit maximally monotone extensions of monotone linear relations that is, A w, B u + B w, A u = w, (AB + BA )u = 0 ∀u ∈ V+ (AB ). (4.12) Because AB +BA : V+ (AB ) → V+ (AB ) is onto by Proposition 4.1.1(i), we obtain that (4.12) holds if and only if w ∈ V− (AB ) + V0 (AB ). Hence (gra J)⊥ = {(A w, B w) | w ∈ V− (BA ) + V0 (BA )}, from which gra J ∗ = gra G. Then by (i), gra G = gra(G)∗ = gra J ∗∗ = gra J. We are ready to apply the Brezis-Browder Theorem, namely Fact 3.2.13, to improve Crouzeix and Oca˜ na-Anaya’s characterizations of monotonicity and maximal monotonicity of G and provide a different proof. Theorem 4.2.3 Let G, G be defined in Lemma 4.2.2. The following are equivalent: (i) G is monotone; (ii) G is monotone; (iii) G is maximally monotone; (iv) G is maximally monotone; 70 4.2. Explicit maximally monotone extensions of monotone linear relations (v) dim V+ (BA ) = p − n, equivalently, AB + BA has exactly p − n positive eigenvalues (counting multiplicity). Proof. (i)⇔(ii): Lemma 4.2.2(iii) and Corollary 4.1.2(i). (ii)⇔(iii)⇔(iv): Note that G = G ∗ and G is always a monotone linear relation by Corollary 4.1.2(iv). It suffices to combine Lemma 4.2.2 and Fact 3.2.13. (i)⇒(v): Assume that G is monotone. Then G is monotone by Lemma 4.2.2(iii) and Corollary 4.1.2(i). By Lemma 4.2.2(ii), Corollary 4.1.2(iv) and Fact 3.2.13, G is maximally monotone, so that dim(gra G) = p−k = n by Fact 3.2.6 and Lemma 4.1.3, thus k = p−n. Note that for each eigenvalue of a symmetric matrix, its geometric multiplicity is the same as its algebraic multiplicity [59, page 512]. (v)⇒(i): Assume that k = p − n. Then dim(gra G) = p − k = n by Lemma 4.1.3, so that G is maximally monotone by Fact 3.2.6(i)(ii). By Lemma 4.2.2(ii) and Fact 3.2.13, G is monotone, which implies that G is monotone. Corollary 4.2.4 Assume that G is monotone. Then B gra G = gra G + u u ∈ V+ (BA A = {(x, x∗ ) | Vg Ax + Vg Bx∗ = 0} 71 4.2. Explicit maximally monotone extensions of monotone linear relations is a maximally monotone extension of G, where v p−n+1 v p−n+2 Vg = . . . . vp Proof. Combine Theorem 4.2.3 and Lemma 4.2.2(iii) directly. Note that Corollary 4.2.4 gives both types of maximally monotone extensions of G, namely, type (4.9) and type (4.10). A remark is in order to compare our extension with the one by Crouzeix and Oca˜ na-Anaya. Remark 4.2.5 (i). Crouzeix and Oca˜ na-Anaya [39] defines the union of monotone extension of G as B S = gra G + u u ∈ K , A where K = {u ∈ Rn | u, (AB + BA )u ≥ 0}. Although this is the set monotonically related to G, it is not monotone in general as long as (AB + BA ) has both positive eigenvalues and negative eigenvalues. Indeed, let (α1 , u1 ) and (α2 , u2 ) be eigen-pairs of (AB + BA ) with α1 > 0 and α2 < 0. We have u1 , (AB +BA )u1 = α1 u1 2 > 0, u2 , (AB +BA )u2 = α2 u2 2 < 0. 72 4.2. Explicit maximally monotone extensions of monotone linear relations Choose > 0 sufficiently small so that u1 + u2 , (AB + BA )(u1 + u2 ) > 0. Then However, has B B u1 , (u1 + u2 ) ∈ S. A A B B B (u1 + u2 ) − u1 = u2 A A A B u2 , A u2 = 2 u2 , BA u2 = 2 u2 , (AB + BA )u2 < 0. 2 Therefore S is not monotone. By using V+ (BA ) ⊆ K, we have obtained a maximally monotone extension of G . (ii). Crouzeix and Oca˜ na-Anaya [39] find a maximally monotone linear subspace extension of G algorithmically by using u ˜k ∈ gra Gk \ gra Gk and constructing gra Gk+1 = gra Gk + R˜ uk where Bk u ˜k = uk , Ak uk , (Ak Bk + Bk Ak )uk ≥ 0. This recursion is done until dim gra Gk = n. In particular, each uk may be chosen as an eigenvector associated with a positive eigenvalue of Ak Bk + Bk Ak , which is possible since p > n when Gk is not maximally monotone. 73 4.2. Explicit maximally monotone extensions of monotone linear relations Their construction uses both formulations, namely, (4.9) and (4.10). No computation indications are given on the passage from one formulation to the other one. The following result extends the characterization of maximally monotone linear relations given by Crouzeix and Oca˜ na-Anaya [39]. Theorem 4.2.6 Let G, G be defined in Lemma 4.2.2. The following are equivalent: (i) G is maximally monotone; (ii) p = n and G is monotone; (iii) p = n and AB + BA is negative semidefinite. (iv) p = n and G is maximally monotone. Proof. (i)⇒(ii): Apply Lemma 4.1.6(iii). (ii)⇒(iii): Apply Theorem 4.2.3(i)(v) directly . (iii)⇒(i): Assume that p = n and (AB + BA ) is negative semidefinite. Then k = 0 and G = G. It follows that dim(gra G) = p − k = n by Lemma 4.1.3, so that G is maximally monotone by Corollary 4.1.2(iv) and Fact 3.2.6(i)(ii). Since G ∗ = G by Lemma 4.2.2(ii), Fact 3.2.13 gives that G = G is maximally monotone. (iii)⇒(iv): Assume that p = n and (AB +BA ) is negative semidefinite. We have k = 0 and dim(gra G) = p − k = n − 0 = n. Hence (iv) holds by Corollary 4.1.2(iv) and Fact 3.2.6(i)(ii). 74 4.2. Explicit maximally monotone extensions of monotone linear relations (iv)⇒(iii): Assume that G is maximally monotone and p = n. We have dim(gra G) = p − k = n − k = n so that k = 0. Hence (AB + BA ) is negative semidefinite. Corollary 4.2.4 supplies only one maximally monotone linear subspace extension of G. Can we find all of them? Surprisingly, we may give a characterization of all the maximally monotone linear subspace extensions of G when it is given in the form of (4.9). Theorem 4.2.7 Let G be monotone. Then G is a maximally monotone extension of G if and only if there exists N ∈ Rp×p with rank of n such that N Idλ N is negative semidefinite and gra G = {(x, x∗ ) | N V Ax + N V Bx∗ = 0}. (4.13) Proof. “⇒”: By Lemma 4.1.6(i), we have gra G∗ = {(B u, −A u) | u ∈ Rp }. (4.14) Since gra G ⊆ gra G and thus gra(G)∗ is a subspace of gra G∗ . Thus by (4.14), there exists a subspace F of Rp such that gra(G)∗ = {(B u, −A u) | u ∈ F }. (4.15) By Fact 3.2.13, Fact 3.2.6 and Lemma 4.1.3, we have dim F = n. (4.16) 75 4.2. Explicit maximally monotone extensions of monotone linear relations Thus, there exists N ∈ Rp×p with rank n such that ran V N = F and gra(G)∗ = {(B V N y, −A V N y) | y ∈ Rp }. (4.17) As G is maximally monotone, (G)∗ is maximally monotone by Fact 3.2.13, so N V (BA + AB )V N is negative semidefinite. Using Fact 4.1.4, we have N Idλ N = N V V Idλ N = N V (AB + BA )V N (4.18) which is negative semidefinite. (4.13) follows from (4.17) by Lemma 4.2.1 using M = V N . “⇐”: By Lemma 4.2.1, we have gra(G)∗ = {(B V N u, −A V N u) | u ∈ Rp }. (4.19) Observe that (G)∗ is monotone because N V (AB + BA )V N = N Idλ N is negative semidefinite by Fact 4.1.4 and the assumption. As rank(V N ) = n, it follows from (4.19) and Lemma 4.1.3 that dim gra(G)∗ = n. Therefore (G)∗ is maximally monotone by Fact 3.2.6. Applying Fact 3.2.13 for T = (G)∗ yields that G = (G)∗∗ is maximally monotone. From the above proof, we see that to find a maximally monotone extension of G one essentially need to find subspace F ⊆ Rp such that dim F = n and AB + BA is negative semidefinite on F . If F = ran M and M ∈ Rp×p 76 4.2. Explicit maximally monotone extensions of monotone linear relations with rank M = n, one can let N = V M . The maximally monotone linear subspace extension of G is G = {(x, x∗ ) | M Ax + M Bx∗ = 0}. In Corollary 4.2.4, one can choose M = 0 0 · · · 0 n vp−n+1 · · · vp . Corollary 4.2.8 Let G be monotone. Then G is a maximally monotone extension of G if and only if there exists M ∈ Rp×p with rank of n such that M (AB + BA )M is negative semidefinite and gra G = {(x, x∗ ) | M Ax + M Bx∗ = 0}. (4.20) Note that G may have different representations in terms of A, B. The maximally monotone extension of G given in Theorem 4.2.7 and Corollary 4.2.4 relies on A, B matrices and N . This might lead to different maximally monotone extensions, see Section 4.5. Remark 4.2.9 A referee for the paper [88] pointed out that there is a shorter way to see Theorem 4.2.7. Consider the maximally monotone linear subspace extension of G of type: gra G = {(x, x∗ ) ∈ Rn × Rn | Ax + Bx∗ = 0} ⊇ gra G where A, B ∈ Rn×n . With the nonsingular p × p matrix V given as in (4.6), 77 4.2. Explicit maximally monotone extensions of monotone linear relations an equivalent formulation of G is gra G = {(x, x∗ ) ∈ Rn × Rn | V Ax + V Bx∗ = 0}. As G is maximally monotone, the n × 2n matrix has rank(A, B) = n and the matrix AB + B A ∈ Rn×n is negative semidefinite. Since gra G ⊇ gra G, we have A (V A) ran = (gra G)⊥ ⊆ (gra G)⊥ = ran . B (V B) Therefore, there exists a p × n matrix N with rank N = n such that A (V A) (V A) N = N = B (V B) (V B) N from which A = N V A, B = N V B. Then the n × n matrix AB + B A = N V A(N V B) + N V B(N V A) (4.21) = N V (AB + BA )V N (4.22) = N Idλ N. (4.23) Therefore, all maximally monotone linear subspace extensions of G can be 78 4.3. Minty parameterizations obtained by using gra G = {(x, x∗ ) ∈ Rn × Rn | N V Ax + N V Bx∗ = 0} in which the p × n matrix N satisfies rank N = n and N Idλ N is negative semidefinite. 4.3 Minty parameterizations Although G is set-valued in general, when G is monotone it has an elegant Minty parametrization in terms of A, B, which is what we are going to show in this section. Lemma 4.3.1 The linear relation G is monotone if and only if 2 − y∗ 2 ≥ 0, whenever (4.24) (A + B)y + (B − A)y ∗ = 0. (4.25) y Consequently, if G is monotone then the p × n matrix B − A must have full column rank, namely n. Proof. Define the 2n × 2n matrix 0 Id P = Id 0 79 4.3. Minty parameterizations where Id ∈ Rn×n . It is easy to see that G is monotone if and only if x (x, x∗ ), P x∗ ≥ 0, whenever Ax + Bx∗ = 0. Define the orthogonal matrix 1 Id − Id Q= √ 2 Id Id and put x y = Q . x∗ y∗ Then G is monotone if and only if y 2 − y∗ 2 ≥ 0, whenever (A + B)y + (B − A)y ∗ = 0. (4.26) (4.27) If (B − A) does not have full column rank, then there exists y ∗ = 0 such that (B − A)y ∗ = 0. Then (0, y ∗ ) satisfies (4.27) but (4.26) fails. Therefore, B − A has to be full column rank. Theorem 4.3.2 (Minty parametrization) Assume that G is a monotone operator. Then (x, x∗ ) ∈ gra G if and only if 1 [Id +(B − A)† (B + A)]y 2 1 x∗ = [Id −(B − A)† (B + A)]y 2 x= (4.28) (4.29) 80 4.3. Minty parameterizations for y = x + x∗ ∈ ran(Id +G). Here the Moore-Penrose inverse (B − A)† = [(B −A) (B −A)]−1 (B −A) . In particular, when G is maximally monotone, we have gra G = {((B − A)−1 By, −(B − A)−1 Ay) | y ∈ Rn }. Proof. As (B − A) is full column rank, (B − A) (B − A) is invertible. It follows from (4.25) that (B − A) (A + B)y + (B − A) (B − A)y ∗ = 0 so that y ∗ = −((B − A) (B − A))−1 (B − A) (A + B)y = −(B − A)† (A + B)y. Then 1 x = √ (y − y ∗ ) = 2 1 x∗ = √ (y + y ∗ ) = 2 where y = ∗ x+x √ 2 1 √ [Id +(B − A)† (B + A)]y 2 1 √ [Id −(B − A)† (B + A)]y 2 with (x, x∗ ) ∈ gra G. Since ran(Id +G) is a subspace, we have 1 [Id +(B − A)† (B + A)]˜ y 2 1 y x∗ = [Id −(B − A)† (B + A)]˜ 2 x= with y˜ = x + x∗ ∈ ran(Id +G). If G is maximally monotone, then p = n by Theorem 4.2.6 and hence B −A is invertible, thus (B −A)† = (B −A)−1 . Moreover, ran(G+Id) = Rn . 81 4.3. Minty parameterizations Then (4.28) and (4.29) imply that 1 x = (B − A)−1 [B − A + (B + A)]y = (B − A)−1 By 2 1 x∗ = (B − A)−1 [(B − A) − (B + A)]y = −(B − A)−1 Ay 2 (4.30) (4.31) for y ∈ Rn . Remark 4.3.3 See Lemma 4.1.5 for ran(G + Id). Note that as G is a monotone linear relation, the mapping z → ((G + Id)−1 , Id −(G + Id)−1 )(z) is bijective and linear from ran(G+Id) to gra G, therefore dim(ran(G+Id)) = dim(gra G). Corollary 4.3.4 Let G be a monotone operator. Then G defined in Corollary 4.2.4, the maximally monotone extension of G, has its Minty parametrization given by gra G = {((Vg B − Vg A)−1 Vg By, −(Vg B − Vg A)−1 Vg Ay) | y ∈ Rn } where Vg is given as in Corollary 4.2.4. Proof. Since rank(Vg ) = n and rank(A B) = p, by Lemma 4.1.3(4.7), rank(Vg A Vg B) = n. Then we can apply Corollary 4.2.4 and Theorem 4.3.2 directly. 82 4.4. Maximally monotone extensions with the same domain or the same range Corollary 4.3.5 When G is maximally monotone, dom G = (B − A)−1 (ran B), ran G = (B − A)−1 (ran A). Recall that T : Rn → Rn is firmly nonexpansive if Tx − Ty 2 ≤ T x − T y, x − y ∀ x, y ∈ dom T. In terms of matrices, we have Corollary 4.3.6 Suppose that p = n, AB + BA is negative semidefinite. Then (B − A)−1 B and −(B − A)−1 A are firmly nonexpansive. Proof. By Theorem 4.2.6, G is maximally monotone. Theorem 4.3.2 gives that (B − A)−1 B = (Id +G)−1 , −(B − A)−1 A = (Id +G−1 )−1 . Being resolvents of monotone operators G, G−1 , they are firmly nonexpansive, see [9, 43] or [13, Fact 2.5]. 4.4 Maximally monotone extensions with the same domain or the same range How do we find maximally monotone linear subspace extensions of G if it is given in the form of (4.10)? The purpose of this section is to find maximally monotone linear subspace extensions of G which keep either dom G or ran G 83 4.4. Maximally monotone extensions with the same domain or the same range unchanged. For a closed convex set S ⊆ Rn , let NS denote its normal cone mapping. Proposition 4.4.1 Assume that T : Rn ⇒ Rn is a monotone linear relation. Then (i) T1 = T + Ndom T , i.e., x → T1 x = T x + (dom T )⊥ ∅ if x ∈ dom T otherwise is maximally monotone. In particular, dom T1 = dom T . (ii) T2 = (T −1 + Nran T )−1 is a maximally monotone extension of T and ran T2 = ran T . Proof. (i): Since 0 ∈ T 0 ⊆ (dom T )⊥ by [15, Proposition 2.2(i)], we have T1 0 = T 0+(dom T )⊥ = (dom T )⊥ so that dom T1 = dom T = (T1 0)⊥ . Hence T1 is maximally monotone by Fact 3.2.6. (ii): Apply (i) to T −1 to see that T −1 + Nran T is a maximally monotone extension of T −1 with dom(T −1 + Nran T ) = ran T . Therefore, T2 is a maximally monotone extension of T with ran T2 = ran T . Define linear relations Ei : Rn ⇒ Rn (i = 1, 2) by Cy 0 2n−p gra E1 = + y ∈ R , ⊥ Dy (ran C) (4.32) 84 4.4. Maximally monotone extensions with the same domain or the same range ⊥ Cy (ran D) 2n−p gra E2 = + y ∈ R . Dy 0 Theorem 4.4.2 (4.33) (i) E1 is a maximally monotone extension of G with dom E1 = dom G. Moreover, 0 C C 0 gra E1 = ran + = ran + . (4.34) D (ran C)⊥ D ker C (ii) E2 is a maximally monotone extension of G with ran E2 = ran G. Moreover, ⊥ C (ran D) C ker D gra E2 = ran + = ran + . (4.35) D D 0 0 Proof. (i): Note that dom G = ran C. The maximal monotonicity follows from Proposition 4.4.1. (4.34) follows from (4.32) and the fact that (ran C)⊥ = ker C [59, page 405]. (ii): Apply (i) to G−1 , i.e., gra G−1 Dy 2n−p = y∈R Cy (4.36) and followed by taking the set-valued inverse. Apparently, both extensions E1 , E2 rely on gra G, dom G, ran G, not on the A, B. In this sense, E1 , E2 are intrinsic maximally monotone linear subspace extensions. 85 4.4. Maximally monotone extensions with the same domain or the same range Remark 4.4.3 Theorem 4.4.2 is much easier to use than Corollary 4.2.8 when G is written in the form of (4.10). Indeed, it is not hard to check that gra(E1∗ ) = {(B u, −A u) | B u ∈ dom G, u ∈ Rp }. (4.37) gra(E2∗ ) = {B u, −A u) | A u ∈ ran G, u ∈ Rp }. (4.38) According to Fact 3.2.13, Ei∗ is maximally monotone and dim Ei∗ = n. This implies that dim{u ∈ Rp | B u ∈ dom G} = n, dim{u ∈ Rp | A u ∈ ran G} = n. Let Mi ∈ Rp×p with rank Mi = n and {u ∈ Rp | B u ∈ dom G} = ran M1 , (4.39) {u ∈ Rp | A u ∈ ran G} = ran M2 . (4.40) Corollary 4.2.8 shows that gra Ei = {(x, x∗ ) | Mi Ax + Mi Bx∗ = 0}. However, finding Mi from (4.39) and (4.40) may not be as easy as it seems. Remark 4.4.4 Unfortunately, we do not know how to determine all maximally monotone linear subspace extensions of G if it is given in the form of 86 4.5. Examples (4.10). 4.5 Examples In the final section, we illustrate our maximally monotone extensions by considering three examples. In particular, they show that maximally monotone extensions G rely on the representation of G in terms of A, B and choices of N we shall use. However, the maximally monotone extensions Ei are intrinsic, depending only on gra G. Example 4.5.1 Consider gra G = (x, x∗ ) ∈ Rn × Rn 0 ∗ Id x+ x = 0 C 0 where C ∈ Rn×n is symmetric and positive definite, and Id ∈ Rn×n . Clearly, 0 gra G = . 0 We have (i) For every α ∈ [−1, 1] , Gα defined by gra Gα = {(0, Rn )} , (x, 1+α C −1 x) | x ∈ Rn , 1−α if α = 1; otherwise is a maximally monotone linear extension of G. 87 4.5. Examples (ii) E1 = G1 and E2 = G−1 . Proof. (i): To find Gα , we need eigenvectors of 0 C 0 Id A = (0 C ) + (Id 0) = . C 0 C 0 Counting multiplicity, the positive definite matrix C has eigen-pairs (λi , wi ) (i = 1, . . . , n) such that λi > 0, wi = 1 and wi , wj = 0 for i = j. As such, the matrix A has 2n eigen-pairs, namely wi λi , wi and wi −λi , −wi with i = 1, . . . , n. Put W = (w1 · · · wn ) ∈ Rn×n and write W V = W W . −W Then W CW = D = diag(λ1 , λ2 , . . . , λn ). 88 4.5. Examples In Theorem 4.2.7, take 0 α Id 2n×2n Nα = ∈R 0 Id where Id ∈ Rn×n . We have rank Nα = n, 0 0 0 0 Nα Idλ Nα = = 0 (α2 − 1)D 0 (α2 − 1)W CW being negative semidefinite, and 0 (1 + α)W V Nα = . 0 (α − 1)W Then by Theorem 4.2.7, we have a maximally monotone linear extension Gα given by gra Gα = 0 (x, x∗ ) ∈ Rn × Rn =0 (1 + α)W x + (α − 1)W Cx∗ = (x, x∗ ) ∈ Rn × Rn | (1 + α)x + (α − 1)Cx∗ = 0 {(0, Rn )} , if α = 1; = (x, 1+α C −1 x) | x ∈ Rn , otherwise. 1−α Hence we get the desired result. (ii): It is immediate from Theorem 4.4.2 and (i). 89 4.5. Examples Example 4.5.2 Consider −1 0 1 0 ∗ x x 1 1 ∗ 2 2 + . gra G = (x, x ) ∈ R × R 0 = 0 0 0 1 x2 x∗2 0 −1 0 1 Then (i) the linear operators Gi : R2 ⇒ R2 for i = 1, 2 given by 1 G1 = 0 0 √ , −1+√ 2 2− 2 1 G2 = 0 2 5 √ 2 10 are two maximally monotone extensions of G. (ii) E1 (x1 , 0) = (x1 , R) ∀x1 ∈ R. (iii) E2 (x1 , y) = (x1 , 0) Proof. We have ∀x1 , y ∈ R. x 1 0 gra G = x1 ∈ R x1 0 is monotone. Since dim G = 1, G is not maximally monotone by Fact 3.2.6. 90 4.5. Examples The matrix 0 −2 0 AB + BA = 0 0 −1 0 −1 −2 has a positive eigenvalue −1 + 0 u= 1 √ 1− 2 √ 2 with an eigenvector 0 √ 2− 2 B so that u = . A 0 √ −1 + 2 Then by Corollary 4.2.4, x 0 1 √ 0 2 − 2 gra G1 = x1 ∈ R + x2 x2 ∈ R 0 x1 √ 0 −1 + 2 x 1 √ (2 − 2)x2 = x1 , x2 ∈ R . x1 √ (−1 + 2)x 2 Therefore, 1 G1 = 0 0 √ −1+√ 2 2− 2 is a maximally monotone extension of G. 91 4.5. Examples Now we have √ 0 0 −1 + 2 √ Idλ = 0 −1 − 2 0 , 0 0 −2 0 1 V = − −1+√2 1 0 − −1−1 √2 1 1 0 . 0 (4.41) Take 0 −1 1 . N = 0 2 −1 0 1 1 (4.42) We have rank N = 2 and 0 0 0 √ √ N Idλ N = 0 −7 − 3 2 1 + 2 √ 0 1+ 2 −4 (4.43) is negative semidefinite. By Theorem 4.2.7, with V, N given in (4.41) and (4.42), we use the NullSpace command in Maple to solve (V N ) Ax + (V N ) Bx∗ = 0, 92 4.5. Examples and get gra G2 = span √ −1 1 1 −2 2 Thus G2 = √ = 0 5 2 0 extension of G. 1 0 , 1 0 2 5 √ 2 10 √ −2 2 √ 5 2 . 0 1 is another maximally monotone On the other hand, gives x 0 x 1 1 0 0 0 gra E1 = x1 ∈ R + = x1 ∈ R x1 0 x1 0 R R E1 (x1 , 0) = (x1 , R) ∀x1 ∈ R. We have x 1 R gra E2 = x1 ∈ R , x1 0 which gives E2 (x1 , y) = (x1 , 0) ∀x1 , y ∈ R. 93 4.5. Examples In [11], the authors use autoconjugates to find maximally monotone extensions of monotone operators. In general, it is not clear whether the maximally monotone extensions of a linear relation is still a linear relation. As both monotone operators in Examples 4.5.2 and 4.5.1 are subsets of {(x, x) | x ∈ Rn }, [11, Example 5.10] shows that the maximally monotone extension obtained by autoconjugates must be Id, which is different from the ones given here. Example 4.5.3 Consider gra G = {(x, x∗ ) ∈ R2 × R2 | Ax + Bx∗ = 0} where 1 1 1 5 1 1 1 5 A= 2 0 , B = 1 7 , thus (A B) = 2 0 1 7 . 3 1 0 2 3 1 0 2 (4.44) Then the linear operators Gi : Rn ⇒ Rn for i = 1, 2 given by G1 = √ −117+17 √ 201 2(−1+ 201) √ −23+3√ 201 − 2(−1+ 201) √ −107+7 √ 201 2(−1+ 201) , √ −21+√201 − 2(−1+ 201) G2 = √ 201 − 6 √ 201 − 29 20 + 30 33 4 √ − 201 6 √ 201 9 − 20 + 30 13 4 are two maximally monotone linear extensions of G. Moreover, −1 0 1 0 gra E1 = x1 + x2 x1 , x2 ∈ R −5 1 1 1 94 4.5. Examples and −1 1 1 5 gra E2 = x1 + x2 x1 , x2 ∈ R . −5 0 1 0 Proof. We have rank(A B) = 3 and √ 0 13 + 201 0 , Idλ = 0 −6 0 √ 0 0 13 − 201 20 √ 1+ 201 V = 0 1 −1 1 1 20 √ 1− 201 1 1 , (4.45) and Vg = 0 20 √ 1− 201 −1 1 . 1 1 (4.46) Clearly, here p = 3, n = 2 and AB + BA has exactly p − n = 3 − 2 = 1 positive eigenvalue. By Theorem 4.2.3(i)(v), G is monotone. Since AB + BA is not negative semidefinite, by Theorem 4.2.6(i)(iii), G is not maximally monotone. With Vg given in (4.46) and A, B in (4.44), use the NullSpace command 95 4.5. Examples in maple to solve Vg Ax + Vg Bx∗ = 0 and obtain G1 defined by √ √ −107+7 −21+√201 √ 201 − − 2(−1+√ 201) 2(−1+ √201) −23+3√ 201 −117+17 √ 201 2(−1+ 201) 2(−1+ 201) gra G1 = span , . 1 0 0 1 By Corollary 4.2.4, G1 is a maximally monotone linear subspace extension of G. Then √ −21+√201 − G1 = 2(−1+√ 201) −23+3√ 201 2(−1+ 201) −1 √ √ 201 − −107+7 2(−1+ 201) √ −117+17 √ 201 2(−1+ 201) = √ −117+17 √ 201 2(−1+ 201) √ −23+3√ 201 − 2(−1+ 201) √ −107+7 √ 201 2(−1+ 201) . √ −21+√201 − 2(−1+ 201) Let N be defined by 1 5 0 0 N = 0 1 0 . 0 0 1 (4.47) Then rank N = 2 and 0 0 N Idλ N = 0 −6 0 0 0 0 √ 338−24 201 25 is negative semidefinite. With N in (4.47), A, B in (4.44) and V in (4.45), use the NullSpace command in maple to solve (V N ) Ax + (V N ) Bx∗ = 0. By Theorem 4.2.7, 96 4.5. Examples we get a maximally monotone linear extension of G, G2 , defined by √ 201 9 − 20 + 30 G2 = √ 29 201 − 20 30 −1 √ + 201 6 √ 201 33 − 4 6 − 13 4 = 33 4 − √ − 29 20 + 201 6 √ 201 30 13 4 − √ 9 − 20 + 201 6 . √ 201 30 To find E E2 , using the LinearSolve command in Maple, we get 1 and C gra G = ran , where D −1 C = , 1 −5 D = . 1 It follows from Theorem 4.4.2 that and −1 0 0 1 gra E1 = x1 + x2 x1 , x2 ∈ R −5 1 1 1 −1 1 5 1 gra E2 = x1 + x2 x1 , x2 ∈ R . −5 0 1 0 97 4.6. Discussion 4.6 Discussion A direction for future work in this chapter is to write computer code to find the maximally monotone subspace extension of G, and to generalize the results into a Hilbert space by applying the Brezis-Browder Theorem. 98 Chapter 5 The sum problem Let A and B be maximally monotone operators from X to X ∗ . Clearly, the sum operator A + B : X ⇒ X ∗ : x → Ax + Bx = a∗ + b∗ | a∗ ∈ Ax and b∗ ∈ Bx is monotone. Rockafellar established the following very important result in 1970. Theorem 5.0.1 (Rockafellar’s sum theorem) (See [66, Theorem 1].) Suppose that X is reflexive. Let A, B : X ⇒ X ∗ be maximally monotone. Assume that A and B satisfy the classical constraint qualification dom A ∩ int dom B = ∅ Then A + B is maximally monotone. The most famous open problem concerns the maximal monotonicity of the sum of two maximally monotone operators in general Banach spaces, which is called the “sum problem”. See Simons’ monograph [74] and [22–24, 86, 90] for a comprehensive account of some recent developments. In this chapter, we prove the maximal monotonicity of A + B provided that dom A ∩ int dom B = ∅, A + Ndom B is of type (FPV), and dom A ∩ dom B ⊆ dom B. 99 5.1. Basic properties We also show the maximal monotonicity of A + B when A is a maximally monotone linear relation and B is a subdifferential operator satisfying dom A ∩ int dom B = ∅. This chapter is mainly based on my work in [90, 91]. 5.1 Basic properties Fact 5.1.1 (Rockafellar) (See [65, Theorem 3], [74, Corollary 10.3 and Theorem 18.1], or [92, Theorem 2.8.7(iii)].) Let f, g : X → ]−∞, +∞] be proper convex functions. Assume that there exists a point x0 ∈ dom f ∩dom g such that g is continuous at x0 . Then for every z ∗ ∈ X ∗ , there exists y ∗ ∈ X ∗ such that (f + g)∗ (z ∗ ) = f ∗ (y ∗ ) + g∗ (z ∗ − y ∗ ). (5.1) Furthermore, ∂(f + g) = ∂f + ∂g. Fact 5.1.2 (Rockafellar) (See [67, Theorem A], [92, Theorem 3.2.8], [74, Theorem 18.7] or [54, Theorem 2.1]) Let f : X → ]−∞, +∞] be a proper lower semicontinuous convex function. Then ∂f is maximally monotone. Fact 5.1.3 (See [61, Theorem 2.28].) Let A : X ⇒ X ∗ be monotone such that int dom A = ∅. Assume that x ∈ int dom A. Then A is locally bounded at x, i.e., there exist δ > 0 and K > 0 such that sup y ∗ ∈Ay y ∗ ≤ K, ∀y ∈ (x + δBX ) ∩ dom A. Fact 5.1.4 (See [61, Proposition 3.3 and Proposition 1.11].) Let f : X → 100 5.1. Basic properties ]−∞, +∞] be a lower semicontinuous convex function and int dom f = ∅. Then f is continuous on int dom f and ∂f (x) = ∅ for every x ∈ int dom f . Fact 5.1.5 (Fitzpatrick) (See [45, Corollary 3.9].) Let A : X ⇒ X ∗ be maximally monotone, and set FA : X × X ∗ → ]−∞, +∞] : (x, x∗ ) → sup (a,a∗ )∈gra A x, a∗ + a, x∗ − a, a∗ . (5.2) Then for every (x, x∗ ) ∈ X × X ∗ , the inequality x, x∗ ≤ FA (x, x∗ ) is true, and equality holds if and only if (x, x∗ ) ∈ gra A. Fact 5.1.6 (Fitzpatrick) (See [45, Theorem 3.4].) Let A : X ⇒ X ∗ be monotone. Then conv dom A ⊆ PX (dom FA ). Fact 5.1.7 (See [84, Theorem 3.4 and Corollary 5.6] or [74, Theorem 24.1(b)].) Let A, B : X ⇒ X ∗ be maximally monotone operators. Assume λ>0 λ [PX (dom FA ) − PX (dom FB )] is a closed subspace of X. If FA+B ≥ ·, · on X × X ∗, (5.3) then A + B is maximally monotone. Fact 5.1.8 (Simons) (See [74, Theorem 27.1 and Theorem 27.3].) Let A : X ⇒ X ∗ be maximally monotone with int dom A = ∅. Then int dom A = int [PX dom FA ], dom A = PX [dom FA ], and dom A is convex. 101 5.1. Basic properties Fact 5.1.9 (Simons) (See [70, Lemma 2.2].) Let f : X → ]−∞, +∞] be proper, lower semicontinuous and convex. Let x ∈ X and λ ∈ R be such that inf f < λ < f (x) ≤ +∞, and set K := λ − f (a) . a∈X,a=x x − a sup Then K ∈ ]0, +∞[ and for every ε ∈ ]0, 1[, there exists (y, y ∗ ) ∈ gra ∂f such that y − x, y ∗ ≤ −(1 − ε)K y − x < 0. (5.4) Fact 5.1.10 (Simons) (See [74, Theorem 48.6(a)].) Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex. Let (x, x∗ ) ∈ X × X ∗ be such that (x, x∗ ) ∈ / gra ∂f and let α > 0. Then for every ε > 0, there exists (y, y ∗ ) ∈ gra ∂f with y = x and y ∗ = x∗ such that x−y −α <ε x∗ − y ∗ (5.5) x − y, x∗ − y ∗ + 1 < ε. x − y · x∗ − y ∗ (5.6) and Fact 5.1.11 (Simons) (See [74, Corollary 28.2].) Let A : X ⇒ X ∗ be maximally monotone. Then span(PX dom FA ) = span [dom A]. (5.7) 102 5.1. Basic properties Now we cite some results on maximally monotone operators of type (FPV). Fact 5.1.12 (Fitzpatrick-Phelps and Verona-Verona) (See [47, Corollary 3.4], [81, Theorem 3] or [74, Theorem 48.4(d)].) Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex. Then ∂f is of type (FPV). Fact 5.1.13 (Simons) (See [74, Theorem 44.2].) Let A : X ⇒ X ∗ be a maximally monotone of type (FPV). Then dom A = conv dom A = PX dom FA . Fact 5.1.14 (Simons) (See [74, Theorem 46.1].) Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then A is of type (FPV). Fact 5.1.15 (Simons and Verona-Verona) (See [74, Thereom 44.1] or [81].) Let A : X ⇒ X ∗ be maximally monotone. Suppose that for every closed convex subset C of X with dom A ∩ int C = ∅, the operator A + NC is maximally monotone. Then A is of type (FPV). The following statement first appeared in [72, Theorem 41.5]. However, on [74, page 199], concerns were raised about the validity of the proof of [72, Theorem 41.5]. In [85], Voisei recently provided a result that generalizes and confirms [72, Theorem 41.5] and hence the following fact. Fact 5.1.16 (Voisei) Let A : X ⇒ X ∗ be maximally monotone of type (FPV) with convex domain, let C be a nonempty closed convex subset of X, 103 5.1. Basic properties and suppose that dom A ∩ int C = ∅. Then A + NC is maximally monotone. Corollary 5.1.17 Let A : X ⇒ X ∗ be maximally monotone of type (FPV) with convex domain, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Then A + NC is of type (F P V ). Proof. By Fact 5.1.16, A+NC is maximally monotone. Let D be a nonempty closed convex subset of X, and suppose that dom(A + NC ) ∩ int D = ∅. Let x1 ∈ dom A ∩ int C and x2 ∈ dom(A + NC ) ∩ int D. Thus, there exists δ > 0 such that x1 + δUX ⊆ C and x2 + δUX ⊆ D. Then for small enough λ ∈ ]0, 1[, we have x2 + λ(x1 − x2 ) + 12 δUX ⊆ D. Clearly, x2 + λ(x1 − x2 ) + λδUX ⊆ C. Thus x2 + λ(x1 − x2 ) + λδ 2 UX ⊆ C ∩ D. Since dom A is convex, x2 + λ(x1 − x2 ) ∈ dom A and x2 + λ(x1 − x2 ) ∈ dom A ∩ int(C ∩ D). By Fact 5.1.1 , A + NC + ND = A + NC∩D . Then, by Fact 5.1.16 (applied to A and C ∩ D), A + NC + ND = A + NC∩D is maximally monotone. By Fact 5.1.15, A + NC is of type (F P V ). Corollary 5.1.18 Let A : X ⇒ X ∗ be a maximally monotone linear relation, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Then A + NC is of type (F P V ). Proof. Apply Fact 5.1.14 and Corollary 5.1.17. The following Lemma 5.1.19 is from [16, Lemma 2.5]. Lemma 5.1.19 Let C be a nonempty closed convex subset of X such that int C = ∅. Let c0 ∈ int C and suppose that z ∈ X C. Then there exists λ ∈ ]0, 1[ such that λc0 + (1 − λ)z ∈ bdry C. 104 5.1. Basic properties Proof. Let λ = inf t ∈ [0, 1] | tc0 + (1 − t)z ∈ C . Since C is closed, λ = min t ∈ [0, 1] | tc0 + (1 − t)z ∈ C . (5.8) Because z ∈ / C, λ > 0. We now show that λc0 + (1 − λ)z ∈ bdry C. Assume to the contrary that λc0 + (1 − λ)z ∈ int C. Then there exists δ ∈ ]0, λ[ such that λc0 + (1 − λ)z − δ(c0 − z) ∈ C. Hence (λ − δ)c0 + (1 − λ + δ)z ∈ C, which contradicts (5.8). Therefore, λc0 + (1 − λ)z ∈ bdry C. Since c0 ∈ / bdry C, we also have λ < 1. The proof of the next result follows closely the proof of [74, Theorem 53.1]. Lemma 5.1.20 was established by Bauschke, Wang and Yao in [19, Lemma 2.10]. Lemma 5.1.20 Let A : X ⇒ X ∗ be a monotone linear relation, and let f : X → ]−∞, +∞] be a proper lower semicontinuous and convex function. Suppose that dom A ∩ int dom ∂f = ∅, (z, z ∗ ) ∈ X × X ∗ is monotonically related to gra(A + ∂f ) and z ∈ dom A. Then z ∈ dom ∂f . Proof. Let c0 ∈ X and y ∗ ∈ X ∗ be such that c0 ∈ dom A ∩ int dom ∂f and (z, y ∗ ) ∈ gra A. (5.9) Take c∗0 ∈ Ac0 , and set M := max y ∗ , c∗0 , (5.10) 105 5.1. Basic properties D := [c0 , z], and h := f + ιD . By (5.9), Fact 5.1.4 and Fact 5.1.1, ∂h = ∂f + ∂ιD . Set g : X → ]−∞, +∞] : x → h(x + z) − z ∗ , x . It remains to show that 0 ∈ dom ∂g. (5.11) If inf g = g(0), then (5.11) holds. Now suppose that inf g < g(0). Let λ ∈ R be such that inf g < λ < g(0), and set λ − g(x) . x g(x)<λ Kλ := sup (5.12) We claim that Kλ ≤ M. By Fact 5.1.9, we have Kλ ∈ ]0, ∞[ and ∀ε ∈ ]0, 1[, by gra ∂g = gra ∂h − (z, z ∗ ) there exists (x, x∗ ) ∈ gra ∂h such that x − z, x∗ − z ∗ ≤ −(1 − ε)Kλ x − z < 0. (5.13) Since ∂h = ∂f + ∂ιD , there exists t ∈ [0, 1] with x∗1 ∈ ∂f (x) and x∗2 ∈ ∂ιD (x) such that x = tc0 + (1 − t)z and x∗ = x∗1 + x∗2 . Then x − z, x∗2 ≥ 0. Thus, by (5.13), x − z, x∗1 − z ∗ ≤ x − z, x∗1 + x∗2 − z ∗ ≤ −(1 − ε)Kλ x − z < 0. (5.14) 106 5.2. Maximality of the sum of a (FPV) operator and a full domain operator As x = tc0 + (1 − t)z and A is a linear relation, we have (x, tc∗0 + (1 − t)y ∗ ) ∈ gra A. Since (z, z ∗ ) is monotonically related to gra(A + ∂f ), by (5.10), x − z, x∗1 − z ∗ ≥ − x − z, tc∗0 + (1 − t)y ∗ ≥ −M x − z . (5.15) Combining (5.15) and (5.14), we obtain −M x − z ≤ −(1 − ε)Kλ x − z < 0. (5.16) Hence, (1 − ε)Kλ ≤ M . Letting ε ↓ 0, we deduce that Kλ ≤ M . Then, by (5.12) and letting λ ↑ g(0), we get g(y) + M y ≥ g(0), ∀y ∈ X. (5.17) In view of [74, Example 7.1], we conclude that 0 ∈ dom ∂g. Hence (5.11) holds and thus z ∈ dom ∂f . 5.2 Maximality of the sum of a (FPV) operator and a full domain operator The following result plays a key role in the proof of Theorem 5.2.4. The first half of its proof follows along the lines of the proof of [74, Theorem 44.2]. Proposition 5.2.1 Let A, B : X ⇒ X ∗ be maximally monotone with dom A ∩ int dom B = ∅. Assume that A + Ndom B is maximally monotone of type (FPV), and dom A ∩ dom B ⊆ dom B. Then PX [dom FA+B ] = 107 5.2. Maximality of the sum of a (FPV) operator and a full domain operator dom A ∩ dom B. Proof. By Fact 5.1.6, dom A ∩ dom B = dom(A + B) ⊆ PX [dom FA+B ]. It suffices to show that PX [dom FA+B ] ⊆ dom A ∩ dom B. (5.18) After translating the graphs if necessary, we can and do assume that 0 ∈ dom A ∩ int dom B and (0, 0) ∈ gra B. To show (5.18), we take z ∈ PX [dom FA+B ] and we assume to the contrary that z∈ / dom A ∩ dom B. (5.19) Thus α = d(z, dom A ∩ dom B) > 0. Now take y0∗ ∈ X ∗ such that y0∗ = 1 and z, y0∗ ≥ 2 3 z . (5.20) Set Un = [0, z] + α 4n UX , ∀n ∈ N. (5.21) Since 0 ∈ Ndom B (x), ∀x ∈ dom B, gra B ⊆ gra(B + Ndom B ). Since B is maximally monotone and B + Ndom B is a monotone extension of B, we must have B = B + Ndom B . Thus A + B = A + Ndom B + B. (5.22) 108 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Since dom A ∩ dom B ⊆ dom B by assumption, we obtain dom A ∩ dom B ⊆ dom(A + Ndom B ) = dom A ∩ dom B ⊆ dom A ∩ dom B. Hence dom A ∩ dom B = dom(A + Ndom B ). (5.23) By (5.19) and (5.23), z ∈ / dom(A + Ndom B ) and thus (z, ny0∗ ) ∈ / gra(A + Ndom B ), ∀n ∈ N. For every n ∈ N, since z ∈ Un and since A + Ndom B is of type (FPV) by assumption, we deduce the existence of (zn , zn∗ ) ∈ gra(A + Ndom B ) such that zn ∈ Un and z − zn , zn∗ > n z − zn , y0∗ , ∀n ∈ N. (5.24) Hence, using (5.21), there exists λn ∈ [0, 1] such that z − zn − λn z = zn − (1 − λn )z < 41 α, ∀n ∈ N. (5.25) By the triangle inequality, we have z − zn < λn z + 14 α for every n ∈ N. From the definition of α and (5.23), it follows that α ≤ z − zn and hence that α < λn z + 41 α. Thus, 3 4α < λn z , ∀n ∈ N. (5.26) 109 5.2. Maximality of the sum of a (FPV) operator and a full domain operator By (5.25) and (5.20), z − zn − λn z, y0∗ ≥ − zn − (1 − λn )z > − 14 α, ∀n ∈ N. (5.27) ∀n ∈ N. (5.28) By (5.27), (5.20) and (5.26), z − zn , y0∗ > λn z, y0∗ − 41 α > 23 3 4α − 41 α = 14 α, Then, by (5.24) and (5.28), z − zn , zn∗ > 41 nα, By (5.21), there exist tn ∈ [0, 1] and bn ∈ ∀n ∈ N. α 4n UX (5.29) such that zn = tn z + bn . Since tn ∈ [0, 1], there exists a convergent subsequence of (tn )n∈N , which, for convenience, we still denote by (tn )n∈N . Then tn → β, where β ∈ [0, 1]. Since bn → 0, we have zn → βz. (5.30) By (5.23), zn ∈ dom A ∩ dom B; thus, zn − z ≥ α and β ∈ [0, 1[. In view of (5.22) and (5.29), we have, for every z ∗ ∈ X ∗ , FA+B (z, z ∗ ) = FA+Ndom B +B (z, z ∗ ) ≥ ≥ sup {n∈N,y ∗ ∈X ∗ } sup {n∈N,y ∗ ∈X ∗ } [ zn , z ∗ + z − zn , zn∗ + z − zn , y ∗ − ιgra B (zn , y ∗ )] zn , z ∗ + 14 nα + z − zn , y ∗ − ιgra B (zn , y ∗ ) . (5.31) 110 5.2. Maximality of the sum of a (FPV) operator and a full domain operator We now claim that FA+B (z, z ∗ ) = ∞. (5.32) We consider two cases. Case 1 : β = 0. By (5.30) and Fact 5.1.3 (applied to 0 ∈ int dom B), there exist N ∈ N and K > 0 such that Bzn = ∅ and sup y ∗ ∈Bzn y ∗ ≤ K, ∀n ≥ N. (5.33) Then, by (5.31), FA+B (z, z ∗ ) ≥ ≥ sup {n≥N,y ∗ ∈X ∗ } sup {n≥N,y ∗ ∈Bzn } ≥ sup {n≥N } =∞ zn , z ∗ + 14 nα + z − zn , y ∗ − ιgra B (zn , y ∗ ) − zn · z ∗ + 41 nα − z − zn · y ∗ − zn · z ∗ + 14 nα − K z − zn (by (5.33)) (by (5.30)). Thus (5.32) holds. Case 2 : β = 0. Take vn∗ ∈ Bzn . We consider two subcases. Subcase 2.1 : (vn∗ )n∈N is bounded. By (5.31), FA+B (z, z ∗ ) ≥ sup {n∈N} zn , z ∗ + 41 nα + z − zn , vn∗ 111 5.2. Maximality of the sum of a (FPV) operator and a full domain operator ≥ sup − zn · z ∗ + 41 nα − z − zn · vn∗ {n∈N} =∞ (by (5.30) and the boundedness of (vn∗ )n∈N ). Hence (5.32) holds. Subcase 2.2 : (vn∗ )n∈N is unbounded. We first show lim sup z − zn , vn∗ ≥ 0. (5.34) n→∞ Since (vn∗ )n∈N is unbounded and after passing to a subsequence if necessary, we assume that vn∗ = 0, ∀n ∈ N and that vn∗ → +∞. By 0 ∈ int dom B and Fact 5.1.3, there exist δ > 0 and M > 0 such that By = ∅ and sup y ∗ ∈By y ∗ ≤ M, ∀y ∈ δBX . (5.35) Then we have zn − y, vn∗ − y ∗ ≥ 0, ∀y ∈ δUX , y ∗ ∈ By, n ∈ N ⇒ zn , vn∗ − y, vn∗ + zn − y, −y ∗ ≥ 0, ⇒ zn , vn∗ − y, vn∗ ≥ zn − y, y ∗ , ∀y ∈ δUX , y ∗ ∈ By, n ∈ N ∀y ∈ δUX , y ∗ ∈ By, n ∈ N ⇒ zn , vn∗ − y, vn∗ ≥ −( zn + δ)M, ∀y ∈ δUX , n ∈ N (by (5.86)) ⇒ zn , vn∗ ≥ y, vn∗ − ( zn + δ)M, ∀y ∈ δUX , n ∈ N ⇒ zn , vn∗ ≥ δ vn∗ − ( zn + δ)M, ∀n ∈ N ⇒ zn , ∗ vn ∗ vn ≥δ− ( zn +δ)M , ∗ vn ∀n ∈ N. (5.36) 112 5.2. Maximality of the sum of a (FPV) operator and a full domain operator By the Banach-Alaoglu Theorem (see [69, Theorem 3.15]), there exist a weak* convergent subnet (vγ∗ )γ∈Γ of (vn∗ )n∈N , say vγ∗ vγ∗ w* w∗ ∈ X ∗ . (5.37) Using (5.30) and taking the limit in (5.36) along the subnet, we obtain βz, w∗ ≥ δ. (5.38) Since β > 0, we have z, w∗ ≥ δ β > 0. (5.39) Now we assume to the contrary that lim sup z − zn , vn∗ < −ε, n→∞ for some ε > 0. Then, for all n sufficiently large, z − zn , vn∗ < − 2ε , and so z − zn , ∗ vn ∗ vn < −2 ε ∗ vn . (5.40) 113 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Then by (5.30) and (5.37), taking the limit in (5.40) along the subnet again, we see that z − βz, w∗ ≤ 0. Since β < 1, we deduce z, w∗ ≤ 0 which contradicts (5.39). Hence (5.34) holds. By (5.31), FA+B (z, z ∗ ) ≥ sup {n∈N} zn , z ∗ + 14 nα + z − zn , vn∗ ≥ sup − zn · z ∗ + 41 nα + z − zn , vn∗ {n∈N} ≥ lim sup − zn · z ∗ + 14 nα + z − zn , vn∗ n→∞ =∞ (by (5.30) and (5.34)). Hence FA+B (z, z ∗ ) = ∞. (5.41) Therefore, we have proved (5.32) in all cases. However, (5.32) contradicts our original choice that z ∈ PX [dom FA+B ]. Hence PX [dom FA+B ] ⊆ dom A ∩ dom B and thus (5.18) holds. Thus we have PX [dom FA+B ] = dom A ∩ dom B. Corollary 5.2.2 Let A : X ⇒ X ∗ be maximally monotone of type (FPV) with convex domain, and B : X ⇒ X ∗ be maximally monotone with dom A∩ 114 5.2. Maximality of the sum of a (FPV) operator and a full domain operator int dom B = ∅. Assume that dom A ∩ dom B ⊆ dom B. Then PX [dom FA+B ] = dom A ∩ dom B. Proof. Combine Fact 5.1.8, Corollary 5.1.17 and Proposition 5.2.1. Corollary 5.2.3 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and let B : X ⇒ X ∗ be maximally monotone with dom A ∩ int dom B = ∅. Assume that dom A ∩ dom B ⊆ dom B. Then PX [dom FA+B ] = dom A ∩ dom B. Proof. Combine Fact 5.1.8, Corollary 5.1.18 and Proposition 5.2.1. Alternatively, combine Fact 5.1.14 and Corollary 5.2.2. We are now ready for our main result in this section. Theorem 5.2.4 Let A, B : X ⇒ X ∗ be maximally monotone with dom A ∩ int dom B = ∅. Assume that A + Ndom B is maximally monotone of type (FPV), and that dom A ∩ dom B ⊆ dom B. Then A + B is maximally monotone. Proof. After translating the graphs if necessary, we can and do assume that 0 ∈ dom A ∩ int dom B and that (0, 0) ∈ gra A ∩ gra B. By Fact 5.1.5, dom A ⊆ PX (dom FA ) and dom B ⊆ PX (dom FB ). Hence, λ>0 λ PX (dom FA ) − PX (dom FB ) = X. (5.42) 115 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Thus, by Fact 5.1.7, it suffices to show that FA+B (z, z ∗ ) ≥ z, z ∗ , ∀(z, z ∗ ) ∈ X × X ∗ . (5.43) Take (z, z ∗ ) ∈ X × X ∗ . Then FA+B (z, z ∗ ) = sup [ x, z ∗ + z, x∗ − x, x∗ + z − x, y ∗ {x,x∗ ,y ∗ } − ιgra A (x, x∗ ) − ιgra B (x, y ∗ )]. (5.44) Assume to the contrary that FA+B (z, z ∗ ) < z, z ∗ . (5.45) Then (z, z ∗ ) ∈ dom FA+B and, by Proposition 5.2.1, z ∈ dom A ∩ dom B = PX [dom FA+B ]. (5.46) Next, we show that FA+B (λz, λz ∗ ) ≥ λ2 z, z ∗ , ∀λ ∈ ]0, 1[ . (5.47) Let λ ∈ ]0, 1[. By (5.46) and Fact 5.1.8, z ∈ PX dom FB . By Fact 5.1.8 again and 0 ∈ int dom B, 0 ∈ int PX dom FB . Then, by [92, Theorem 1.1.2(ii)], we have λz ∈ int PX dom FB = int [PX dom FB ] . (5.48) 116 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Combining (5.48) and Fact 5.1.8, we see that λz ∈ int dom B. We consider two cases. Case 1 : λz ∈ dom A. By (5.44), FA+B (λz, λz ∗ ) ≥ sup [ λz, λz ∗ + λz, x∗ − λz, x∗ + λz − λz, y ∗ {x∗ ,y ∗ } − ιgra A (λz, x∗ ) − ιgra B (λz, y ∗ )] = λz, λz ∗ . Hence (5.47) holds. Case 2 : λz ∈ / dom A. Using 0 ∈ dom A ∩ dom B and the convexity of dom A ∩ dom B (which follows from (5.46)), we obtain λz ∈ dom A ∩ dom B ⊆ dom A ∩ dom B. Set Un = λz + n1 UX , ∀n ∈ N. (5.49) Then Un ∩ dom(A + Ndom B ) = ∅. Since (λz, λz ∗ ) ∈ / gra(A + Ndom B ), λz ∈ Un , and A + Ndom B is of type (FPV), there exists (bn , b∗n ) ∈ gra(A + Ndom B ) such that bn ∈ Un and λz, b∗n + bn , λz ∗ − bn , b∗n > λ2 z, z ∗ , ∀n ∈ N. (5.50) 117 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Since λz ∈ int dom B and bn → λz, by Fact 5.1.3, there exist N ∈ N and M > 0 such that bn ∈ int dom B and sup v∗ ∈Bbn v ∗ ≤ M, ∀n ≥ N. (5.51) Hence Ndom B (bn ) = {0} and thus (bn , b∗n ) ∈ gra A for every n ≥ N . Thus by (5.44), (5.50) and (5.51), FA+B (λz, λz ∗ ) ≥ ≥ sup {v∗ ∈Bbn } sup {v∗ ∈Bbn } [ bn , λz ∗ + λz, b∗n − bn , b∗n + λz − bn , v ∗ ] , λ2 z, z ∗ + λz − bn , v ∗ ≥ sup λ2 z, z ∗ − M λz − bn ≥ λ2 z, z ∗ , , ∀n ≥ N ∀n ≥ N ∀n ≥ N (by (5.50)) (by (5.51)) (by bn → λz). (5.52) Hence FA+B (λz, λz ∗ ) ≥ λ2 z, z ∗ . We have established that (5.47) holds in both cases. Since (0, 0) ∈ gra A∩ gra B, we obtain (∀(x, x∗ ) ∈ gra(A + B)) x, x∗ ≥ 0. Thus, FA+B (0, 0) = 0. Now define f : [0, 1] → R : t → FA+B (tz, tz ∗ ). Then f is continuous on [0, 1] by [92, Proposition 2.1.6]. From (5.47), we obtain FA+B (z, z ∗ ) = lim FA+B (λz, λz ∗ ) ≥ lim λz, λz ∗ = z, z ∗ , λ→1− λ→1− (5.53) 118 5.2. Maximality of the sum of a (FPV) operator and a full domain operator which contradicts (5.45). Hence FA+B (z, z ∗ ) ≥ z, z ∗ . (5.54) Therefore, (5.43) holds, and A + B is maximally monotone. Theorem 5.2.4 allows us to deduce both new and previously known sum theorems. Corollary 5.2.5 Let f : X → ]−∞, +∞] be proper, lower semicontinuous and convex, and let B : X ⇒ X ∗ be maximally monotone with dom f ∩ int dom B = ∅. Assume that dom ∂f ∩ dom B ⊆ dom B. Then ∂f + B is maximally monotone. Proof. By Fact 5.1.8 and Fact 5.1.1, ∂f + Ndom B = ∂(f + ιdom B ). Then by Fact 5.1.12, ∂f + Ndom B is type of (FPV). Now apply Theorem 5.2.4. Corollary 5.2.6 Let A : X ⇒ X ∗ be maximally monotone of type (FPV), and let B : X ⇒ X ∗ be maximally monotone with full domain. Then A + B is maximally monotone. Proof. Since A + Ndom B = A + NX = A and thus A + Ndom B is maximally monotone of type (FPV), the conclusion follows from Theorem 5.2.4. Corollary 5.2.7 (Verona-Verona) (See [82, Corollary 2.9(a)] or [74, Theorem 53.1].) Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex, and let B : X ⇒ X ∗ be maximally monotone with full domain. Then ∂f + B is maximally monotone. 119 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Proof. Clear from Corollary 5.2.5. Alternatively, combine Fact 5.1.12 and Corollary 5.2.6. Corollary 5.2.8 (Heisler) (See [62, Remark, page 17].) Let A, B : X ⇒ X ∗ be maximally monotone with full domain. Then A + B is maximally monotone. Proof. Let C be a nonempty closed convex subset of X. By Corollary 5.2.7, NC + A is maximally monotone. Thus, A is of type (FPV) by Fact 5.1.15. The conclusion now follows from Corollary 5.2.6. Corollary 5.2.9 Let A : X ⇒ X ∗ be maximally monotone of type (FPV) with convex domain, and let B : X ⇒ X ∗ be maximally monotone with dom A ∩ int dom B = ∅. Assume that dom A ∩ dom B ⊆ dom B. Then A + B is maximally monotone. Proof. Combine Fact 5.1.8, Corollary 5.1.17 and Theorem 5.2.4. Corollary 5.2.10 (Voisei) (See [85].) Let A : X ⇒ X ∗ be maximally monotone of type (FPV) with convex domain, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Then A + NC is maximally monotone. Proof. Apply Corollary 5.2.9. Corollary 5.2.11 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and let B : X ⇒ X ∗ be maximally monotone with dom A∩ int dom B = ∅. Assume that dom A ∩ dom B ⊆ dom B. Then A + B is maximally monotone. 120 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Proof. Combine Fact 5.1.14 and Corollary 5.2.9. Corollary 5.2.12 (See [16, Theorem 3.1].) Let A : X ⇒ X ∗ be a maximally monotone linear relation, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Then A + NC is maximally monotone. Proof. Apply Corollary 5.2.11. Corollary 5.2.13 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and let B : X ⇒ X ∗ be maximally monotone with full domain. Then A + B is maximally monotone. Proof. Apply Corollary 5.2.11. Example 5.2.14 Suppose that X = L1 [0, 1], let D = x ∈ X | x is absolutely continuous, x(0) = 0, x ∈ X ∗ , and set A : X ⇒ X∗ : x → {x }, ∅, if x ∈ D; otherwise. By Phelps and Simons’ [63, Example 4.3], A is an at most single-valued maximally monotone linear relation with proper dense domain, and A is neither symmetric nor skew. Now let J be the duality mapping, i.e., J = ∂ 12 · 2. Then Corollary 5.2.13 implies that A + J is maximally monotone. To the best of our knowledge, the maximal monotonicity of A + J cannot be deduced from any previously known result. 121 5.3. Maximality of the sum of a linear relation and a subdifferential operator Remark 5.2.15 In [19], it was shown that the sum problem has an affirmative solution when A is a linear relation, B is the subdifferential operator of a proper lower semicontinuous sublinear function, and Rockafellar’s constraint qualification holds. When the domain of the subdifferential operator is closed, then that result can be deduced from Theorem 5.2.4. However, it is possible that the domain of the subdifferential operator of a proper lower semicontinuous sublinear function does not have to be closed. For an example, see [3, Example 5.4]: Set C = {(x, y) ∈ R2 | 0 < 1/x ≤ y} and f = ι∗C given by f (x, y) := −2√xy, +∞, if x ≤ 0 and y ≤ 0; otherwise. Then f is not subdifferentiable at any point in the boundary of its domain, except at the origin. Thus, in the general case, we do not know whether or not it is possible to deduce the result in [19] from Theorem 5.2.4. 5.3 Maximality of the sum of a linear relation and a subdifferential operator Theorem 5.3.1 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and let f : X → ]−∞, +∞] be a proper lower semicontinuous convex function with dom A∩int dom ∂f = ∅. Then A+∂f is maximally monotone. Proof. After translating the graphs if necessary, we can and do assume that 0 ∈ dom A ∩ int dom ∂f and that (0, 0) ∈ gra A ∩ gra ∂f . By Fact 5.1.5 and 122 5.3. Maximality of the sum of a linear relation and a subdifferential operator Fact 5.1.2, dom A ⊆ PX (dom FA ) and dom ∂f ⊆ PX (dom F∂f ). Hence, λ>0 λ PX (dom FA ) − PX (dom F∂f ) = X. (5.55) Thus, by Fact 5.1.2 and Fact 5.1.7, it suffices to show that FA+∂f (z, z ∗ ) ≥ z, z ∗ , ∀(z, z ∗ ) ∈ X × X ∗ . (5.56) Take (z, z ∗ ) ∈ X × X ∗ . Then FA+∂f (z, z ∗ ) = sup [ x, z ∗ + z, x∗ − x, x∗ + z − x, y ∗ {x,x∗ ,y ∗ } − ιgra A (x, x∗ ) − ιgra ∂f (x, y ∗ )]. (5.57) Assume to the contrary that FA+∂f (z, z ∗ ) + λ < z, z ∗ , (5.58) (z, z ∗ ) is monotonically related to gra(A + ∂f ). (5.59) where λ > 0. Thus by (5.58), We claim that z∈ / dom A. (5.60) 123 5.3. Maximality of the sum of a linear relation and a subdifferential operator Indeed, if z ∈ dom A, apply (5.59) and Lemma 5.1.20 to get z ∈ dom ∂f . Thus z ∈ dom A∩dom ∂f and hence FA+∂f (z, z ∗ ) ≥ z, z ∗ which contradicts (5.58). This establishes (5.60). By (5.58) and the assumption that (0, 0) ∈ gra A ∩ gra ∂f , we have sup [ 0, z ∗ + z, A0 − 0, A0 + z, ∂f (0) ] = [ z, a∗ + z, b∗ ] < z, z ∗ . sup a∗ ∈A0,b∗ ∈∂f (0) Thus, because A0 is a linear subspace, z ∈ X ∩ (A0)⊥ . (5.61) Then, by Proposition 3.2.2(i), we have z ∈ dom A. (5.62) z ∈ dom A\dom A. (5.63) Combine (5.60) and (5.62), Set Un = z + n1 UX , ∀n ∈ N. (5.64) By (5.63), (z, z ∗ ) ∈ / gra A and Un ∩ dom A = ∅. Since z ∈ Un and A is of type (FPV) by Fact 5.1.14, there exists (an , a∗n ) ∈ gra A with an ∈ Un , n ∈ N 124 5.3. Maximality of the sum of a linear relation and a subdifferential operator such that z, a∗n + an , z ∗ − an , a∗n > z, z ∗ . (5.65) an → z. (5.66) z ∈ dom ∂f . (5.67) Then we have Now we claim that Suppose to the contrary that z ∈ dom ∂f . By the Brøndsted-Rockafellar Theorem (see [61, Theorem 3.17] or [92, Theorem 3.1.2]), dom ∂f = dom f . Since 0 ∈ int dom ∂f ⊆ int dom f ⊆ int dom f , then by Lemma 5.1.19, there exists δ ∈ ]0, 1[ such that δz ∈ bdry dom f . (5.68) Set gn : X → ]−∞, +∞] by gn = f + ι[0,an ] , n∈N (5.69) Since z ∈ / dom f , z ∈ dom f ∩ [0, an ] = dom gn . Thus (z, z ∗ ) ∈ / gra ∂gn . Then by Fact 5.1.10, there exist βn ∈ [0, 1] and x∗n ∈ ∂gn (βn an ) with x∗n = z ∗ 125 5.3. Maximality of the sum of a linear relation and a subdifferential operator and βn an = z such that z − βn an ≥n z ∗ − x∗n z − βn an , z ∗ − x∗n < − 34 . z − βn an · z ∗ − x∗n (5.70) (5.71) By (5.66), z − βn an is bounded. Then by (5.70), we have x∗n → z ∗ . (5.72) Since 0 ∈ int dom f , f is continuous at 0 by Fact 5.1.4. Then by 0 ∈ dom f ∩ dom ι[0,an ] and Fact 5.1.1, we have that there exist wn∗ ∈ ∂f (βn an ) and vn∗ ∈ ∂ι[0,an ] (βn an ) such that x∗n = wn∗ + vn∗ . Then by (5.72), wn∗ + vn∗ → z ∗ . (5.73) Since βn ∈ [0, 1], there exists a convergent subsequence of (βn )n∈N , which, for convenience, we still denote by (βn )n∈N . Then βn → β, where β ∈ [0, 1]. Then by (5.66), βn an → βz. (5.74) β ≤ δ < 1. (5.75) We claim that 126 5.3. Maximality of the sum of a linear relation and a subdifferential operator In fact, suppose to the contrary that β > δ. By (5.74), βz ∈ dom f . Then by 0 ∈ int dom f and [92, Theorem 1.1.2(ii)], δz = δ β βz ∈ int dom f , which contradicts (5.68). We can and do suppose that βn < 1 for every n ∈ N. Then by vn∗ ∈ ∂ι[0,an ] (βn an ), we have vn∗ , an − βn an ≤ 0. (5.76) Dividing by (1 − βn ) on both sides of the above inequality, we have vn∗ , an ≤ 0. (5.77) Since (0, 0) ∈ gra A, an , a∗n ≥ 0, ∀n ∈ N. Then by (5.65), we have z, βn a∗n + βn an , z ∗ − βn2 an , a∗n ≥ βn z, a∗n + βn an , z ∗ − βn an , a∗n ≥ βn z, z ∗ . (5.78) Then by (5.78), z − βn an , βn a∗n ≥ βn z − βn an , z ∗ . (5.79) Since gra A is a linear subspace and (an , a∗n ) ∈ gra A, (βn an , βn a∗n ) ∈ gra A. By (5.58), we have λ < z − βn an , z ∗ − wn∗ − βn a∗n = z − βn an , z ∗ − wn∗ + z − βn an , −βn a∗n 127 5.3. Maximality of the sum of a linear relation and a subdifferential operator < − 43 z − βn an · z ∗ − wn∗ − vn∗ + z − βn an , vn∗ + z − βn an , −βn a∗n (by (5.71)) ≤ − 34 z − βn an · z ∗ − wn∗ − vn∗ + z − βn an , vn∗ − βn z − βn an , z ∗ (by (5.79)). Then λ < z − βn an , vn∗ − βn z − βn an , z ∗ . (5.80) Now we consider two cases: Case 1 : (wn∗ )n∈N is bounded. By (5.73), (vn∗ )n∈N is bounded. By the Banach-Alaoglu Theorem (see [69, Theorem 3.15]), there exist a weak* convergent subnet (vγ∗ )γ∈Γ of (vn∗ )n∈N , say vγ∗ w* ∗ v∞ ∈ X ∗. (5.81) Combine (5.66), (5.74) and (5.81), and pass the limit along the subnet of (5.80) to get that ∗ λ ≤ z − βz, v∞ . (5.82) 128 5.3. Maximality of the sum of a linear relation and a subdifferential operator By (5.75), divide by (1 − β) on both sides of (5.82) to get ∗ z, v∞ ≥ λ 1−β > 0. (5.83) On the other hand, by (5.66) and (5.81), taking the limit along the subnet of (5.77) we get that ∗ v∞ , z ≤ 0, (5.84) which contradicts (5.83). Case 2 : (wn∗ )n∈N is unbounded. Since (wn∗ )n∈N is unbounded and after passing to a subsequence if necessary, we assume that wn∗ = 0, ∀n ∈ N and that wn∗ → +∞. By the Banach-Alaoglu Theorem again, there exist a weak* convergent subnet (wν∗ )ν∈I of (wn∗ )n∈N , say wν∗ wν∗ w* ∗ w∞ ∈ X ∗. (5.85) By 0 ∈ int dom ∂f and Fact 5.1.3, there exist ρ > 0 and M > 0 such that ∂f (y) = ∅ and sup y ∗ ∈∂f (y) y ∗ ≤ M, ∀y ∈ ρUX . (5.86) Then by wn∗ ∈ ∂f (βn an ), we have βn an − y, wn∗ − y ∗ ≥ 0, ∀y ∈ ρUX , y ∗ ∈ ∂f (y) 129 5.3. Maximality of the sum of a linear relation and a subdifferential operator ⇒ βn an , wn∗ − y, wn∗ + βn an − y, −y ∗ ≥ 0, ⇒ βn an , wn∗ − y, wn∗ ≥ βn an − y, y ∗ , ∀y ∈ ρUX , y ∗ ∈ ∂f (y) ∀y ∈ ρUX , y ∗ ∈ ∂f (y) ⇒ βn an , wn∗ − y, wn∗ ≥ −( βn an + ρ)M, ⇒ βn an , wn∗ ≥ y, wn∗ − ( βn an + ρ)M, ∀y ∈ ρUX (by (5.86)) ∀y ∈ ρUX ⇒ βn an , wn∗ ≥ ρ wn∗ − ( βn an + ρ)M ⇒ βn an , ∗ wn ∗ wn ≥ρ− ( βn an + ρ)M , wn∗ ∀n ∈ N. (5.87) Combining (5.74) and (5.85), taking the limit in (5.87) along the subnet, we obtain ∗ βz, w∞ ≥ ρ. (5.88) Then we have β = 0 and thus β > 0. Then by (5.88), ∗ z, w∞ ≥ By (5.73) and z∗ ∗ wn ρ β > 0. (5.89) → 0, we have wn∗ vn∗ + → 0. wn∗ wn∗ (5.90) By(5.85), taking the weak∗ limit in (5.90) along the subnet, we obtain vν∗ wν∗ w* ∗ −w∞ . (5.91) 130 5.3. Maximality of the sum of a linear relation and a subdifferential operator Dividing by wn∗ on the both sides of (5.80), we get that vn∗ λ < z − β a , n n wn∗ wn∗ − βn z − βn an , z ∗ . wn∗ (5.92) Combining (5.74), (5.66) and (5.91), taking the limit in (5.92) along the subnet, we obtain ∗ z − βz, −w∞ ≥ 0. (5.93) ∗ ≥ 0, z, −w∞ (5.94) By (5.75) and (5.93), which contradicts (5.89). Altogether z ∈ dom ∂f = dom f . Next, we show that FA+∂f (tz, tz ∗ ) ≥ t2 z, z ∗ , ∀t ∈ ]0, 1[ . (5.95) Let t ∈ ]0, 1[. By 0 ∈ int dom f and [92, Theorem 1.1.2(ii)], we have tz ∈ int dom f. (5.96) tz ∈ int dom ∂f. (5.97) By Fact 5.1.4, 131 5.3. Maximality of the sum of a linear relation and a subdifferential operator Set Hn = tz + n1 UX , ∀n ∈ N. (5.98) Since dom A is a linear subspace, tz ∈ dom A\dom A by (5.63). Then Hn ∩ dom A = ∅. Since (tz, tz ∗ ) ∈ / gra A and tz ∈ Hn , A is of type (FPV) by Fact 5.1.14, there exists (bn , b∗n ) ∈ gra A such that bn ∈ Hn and tz, b∗n + bn , tz ∗ − bn , b∗n > t2 z, z ∗ , ∀n ∈ N. (5.99) Since tz ∈ int dom ∂f and bn → tz, by Fact 5.1.3, there exist N ∈ N and K > 0 such that bn ∈ int dom ∂f and sup v∗ ∈∂f (bn ) v ∗ ≤ K, ∀n ≥ N. (5.100) Hence FA+∂f (tz, tz ∗ ) ≥ ≥ sup {c∗ ∈∂f (bn )} sup {c∗ ∈∂f (bn )} [ bn , tz ∗ + tz, b∗n − bn , b∗n + tz − bn , c∗ ] , t2 z, z ∗ + tz − bn , c∗ ≥ sup t2 z, z ∗ − K tz − bn ≥ t2 z, z ∗ (by bn → tz). , , ∀n ≥ N ∀n ≥ N ∀n ≥ N (by (5.99)) (by (5.100)) (5.101) Hence FA+∂f (tz, tz ∗ ) ≥ t2 z, z ∗ . 132 5.3. Maximality of the sum of a linear relation and a subdifferential operator We have established (5.95). Since (0, 0) ∈ gra A ∩ gra ∂f , we obtain (∀(d, d∗ ) ∈ gra(A + ∂f )) d, d∗ ≥ 0. Thus, FA+∂f (0, 0) = 0. Now define j : [0, 1] → R : t → FA+∂f (tz, tz ∗ ). Then j is continuous on [0, 1] by (5.58) and [92, Proposition 2.1.6]. From (5.95), we obtain FA+∂f (z, z ∗ ) = lim FA+∂f (tz, tz ∗ ) ≥ lim tz, tz ∗ = z, z ∗ , t→1− t→1− (5.102) which contradicts (5.58). Hence FA+∂f (z, z ∗ ) ≥ z, z ∗ . (5.103) Therefore, (5.56) holds, and A + ∂f is maximally monotone. Remark 5.3.2 In Theorem 5.3.1, when int dom A ∩ dom ∂f = ∅, we have dom A = X since dom A is a linear subspace. Therefore, we can obtain the maximal monotonicity of A + ∂f from the Verona-Verona result (see [82, Corollary 2.9(a)], [74, Theorem 53.1] or [90, Corollary 3.7]). Corollary 5.3.3 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and f : X → ]−∞, +∞] be a proper lower semicontinuous convex function with dom A ∩ int dom ∂f = ∅. Then A + ∂f is of type (F P V ). Proof. By Theorem 5.3.1, A + ∂f is maximally monotone. Let C be a nonempty closed convex subset of X, and suppose that dom(A+∂f )∩int C = ∅. Let x1 ∈ dom A ∩ int dom ∂f and x2 ∈ dom(A + ∂f ) ∩ int C. Thus, there 133 5.3. Maximality of the sum of a linear relation and a subdifferential operator exists δ > 0 such that x1 + δUX ⊆ dom f and x2 + δUX ⊆ C. Then for small enough λ ∈ ]0, 1[, we have x2 + λ(x1 − x2 ) + 12 δUX ⊆ C. Clearly, x2 +λ(x1 −x2 )+λδUX ⊆ dom f . Thus x2 +λ(x1 −x2 )+ λδ 2 UX ⊆ dom f ∩C = dom(f + ιC ). By Fact 5.1.4, x2 + λ(x1 − x2 ) + λδ 2 UX ⊆ dom ∂(f + ιC ). Since dom A is convex, x2 + λ(x1 − x2 ) ∈ dom A and x2 + λ(x1 − x2 ) ∈ dom A ∩ int [dom ∂(f + ιC )]. By Fact 5.1.1 , ∂f + NC = ∂(f + ιC ). Then, by Theorem 5.3.1 (applied to A and f + ιC ), A + ∂f + NC = A + ∂(f + ιC ) is maximally monotone. By Fact 5.1.15, A + ∂f is of type (F P V ). Corollary 5.3.4 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and f : X → ]−∞, +∞] be a proper lower semicontinuous convex function with dom A ∩ int dom ∂f = ∅. Then dom(A + ∂f ) = conv dom(A + ∂f ) = PX dom FA+∂f . Proof. Combine Corollary 5.3.3 and Fact 5.1.13. Now by Corollary 5.3.3, we can deduce Fact 5.1.14 that is used in the proof of Theorem 5.3.1. Corollary 5.3.5 (Simons) (See [74, Theorem 46.1].) Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then A is of type (FPV). Proof. Let f = ιX . Then by Corollary 5.3.3, we have that A = A + ∂f is of type (FPV). Corollary 5.3.6 (See [16, Theorem 3.1].) Let A : X ⇒ X ∗ be a maximally monotone linear relation, let C be a nonempty closed convex subset of X, 134 5.4. An example and comments and suppose that dom A ∩ int C = ∅. Then A + NC is maximally monotone. Corollary 5.3.7 (See [19, Theorem 3.1].) Let A : X ⇒ X ∗ be a maximally monotone linear relation, let f : X → ]−∞, +∞] be a proper lower semicontinuous sublinear function, and suppose that dom A ∩ int dom ∂f = ∅. Then A + ∂f is maximally monotone. 5.4 An example and comments Example 5.4.1 Suppose that X = L1 [0, 1] with norm · 1, let D = x ∈ X | x is absolutely continuous, x(0) = 0, x ∈ X ∗ , and set A : X ⇒ X∗ : x → Define f : X → ]−∞, +∞] by f (x) = 1 1− x +∞, {x }, ∅, 2 1 , if x ∈ D; otherwise. if x < 1; (5.104) otherwise. Clearly, X is a nonreflexive Banach space. By Phelps and Simons’ [63, Example 4.3], A is an at most single-valued maximally monotone linear relation with proper dense domain, and A is neither symmetric nor skew. Since g(t) = 1 1−t2 is convex and increasing on [0, 1[ (by g (t) = 2(1 − t2 )−2 + 8t2 (1 − t2 )−3 ≥ 0, ∀t ∈ [0, 1[), f is convex. Clearly, f is proper lower 135 5.4. An example and comments semicontinuous, and by Fact 5.1.4, we have dom f = UX = int dom f = dom ∂f = int [dom ∂f ] . (5.105) Since 0 ∈ dom A ∩ int [dom ∂f ], Theorem 5.3.1 implies that A + ∂f is maximally monotone. To the best of our knowledge, the maximal monotonicity of A + ∂f cannot be deduced from any previously known result. Remark 5.4.2 To the best of our knowledge, the results in [19, 82, 84, 86, 90] cannot establish the maximal monotonicity in Example 5.4.1. (1) Verona and Verona (see [82, Corollary 2.9(a)] or [74, Theorem 53.1] or [90, Corollary 3.7]) showed the following: “Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex, let A : X ⇒ X ∗ be maximally monotone, and suppose that dom A = X. Then ∂f + A is maximally monotone.” The dom A in Example 5.4.1 is proper dense, hence A + ∂f in Example 5.4.1 cannot be deduced from the Verona -Verona result. (2) In [84, Theorem 5.10(η)], Voisei showed that the sum problem has an affirmative solution when dom A∩dom B is closed, dom A is convex and Rockafellar’s constraint qualification holds. In Example 5.4.1, dom A ∩ dom ∂f is not closed by (5.105). Hence we cannot apply for [84, Theorem 5.10(η)]. (3) In [86, Corollary 4], Voisei and Z˘ alinescu showed that the sum problem has an affirmative solution when ic (dom A) = ∅,ic (dom B) = ∅ and 0 ∈ic [dom A − dom B]. Since the dom A in Example 5.4.1 is a proper 136 5.4. An example and comments dense linear subspace, ic (dom A) = ∅. Thus we cannot apply for [86, Corollary 4]. (Given a set C ⊆ X, we define ic C= i C, ∅, ic C by if aff C is closed; otherwise, where i C [92] is the intrinsic core or relative algebraic interior of C, defined by i C = {a ∈ C | ∀x ∈ aff(C − C), ∃δ > 0, ∀λ ∈ [0, δ] : a + λx ∈ C}.) (4) In [19], it was shown that the sum problem has an affirmative solution when A is a linear relation, B is the subdifferential operator of a proper lower semicontinuous sublinear function, and Rockafellar’s constraint qualification holds. Clearly, f in Example 5.4.1 is not sublinear. Then we cannot apply for it. Theorem 5.3.1 truly generalizes [19]. (5) In [90, Corollary 3.11], it was shown that the sum problem has an affirmative solution when A is a linear relation, B is a maximally monotone operator satisfying Rockafellar’s constraint qualification and dom A ∩ dom B ⊆ dom B. In Example 5.4.1, since dom A is a linear subspace, we can take x0 ∈ dom A with x0 = 1. Thus, by (5.105), we have that x0 ∈ dom A ∩ UX = dom A ∩ dom ∂f but x0 ∈ UX = dom ∂f. (5.106) Thus dom A ∩ dom ∂f dom ∂f and thus we cannot apply [90, Corol- lary 3.11] either. 137 5.5. Discussion 5.5 Discussion As we can see, Fact 5.1.7 plays an important role in the proof of Theorem 5.2.4 and Theorem 5.3.1. Theorem 5.2.4 presents a powerful sufficient condition for the sum problem. The following question posed by Simons in [72, Problem 41.4] remains open: Let A : X ⇒ X ∗ be maximally monotone of type (FPV), let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Is A + NC necessarily maximally monotone? If the above result holds, by Theorem 5.2.4, we can get the following result: Let A : X ⇒ X ∗ be maximally monotone of type (FPV), and let B : X ⇒ X ∗ be maximally monotone with dom A ∩ int dom B = ∅. Assume that dom A ∩ dom B ⊆ dom B. Then A + B is maximally monotone. 138 Chapter 6 Classical types of maximally monotone operators This chapter is based on the work by Bauschke, Borwein, Wang and Yao in [6, 7]. We study three classical types of maximally monotone operators: dense type, negative-infimum type, and Fitzpatrick-Phelps type. We show that every maximally monotone operator of Fitzpatrick-Phelps type must be of dense type. This provides affirmative answers to two questions posed by Stephen Simons and it implies that various important notions of monotonicity coincide. Moreover, we prove that for a maximally monotone linear relation, the monotonicities of dense type, of negative-infimum type, and of FitzpatrickPhelps type are the same and equivalent to monotonicity of the adjoint. This result also provides an affirmative answer to one problem posed by Phelps and Simons. 6.1 Introduction and auxiliary results We now recall the three fundamental types of monotonicity. 139 6.1. Introduction and auxiliary results Definition 6.1.1 Let A : X ⇒ X ∗ be maximally monotone. Then three key types of monotone operators are defined as follows. (i) A is of dense type or type (D) (1971, [49], [62] and [76, Theorem 9.5]) if for every (x∗∗ , x∗ ) ∈ X ∗∗ × X ∗ with inf (a,a∗ )∈gra A a − x∗∗ , a∗ − x∗ ≥ 0, there exists a bounded net (aα , a∗α )α∈Γ in gra A such that (aα , a∗α )α∈Γ weak*×strong converges to (x∗∗ , x∗ ). (ii) A is of type negative infimum (NI) (1996, [71]) if sup (a,a∗ )∈gra A a, x∗ + a∗ , x∗∗ − a, a∗ ≥ x∗∗ , x∗ , for every (x∗∗ , x∗ ) ∈ X ∗∗ × X ∗ . (iii) A is of type Fitzpatrick-Phelps (FP) (1992, [46]) if whenever U is an open convex subset of X ∗ such that U ∩ ran A = ∅, x∗ ∈ U , and (x, x∗ ) ∈ X × X ∗ is monotonically related to gra A ∩ (X × U ) it must follow that (x, x∗ ) ∈ gra A. All three of these properties are known to hold for the subgradient of a closed convex function and for every maximally monotone operator on a reflexive space [26, 72, 74]. These and other relationships known amongst these and other monotonicity notions are described in [26, Chapter 9]. Now we introduce some notation. Let F : X × X ∗ → ]−∞, +∞]. We say F is a representative of a maximally monotone operator A : X ⇒ X ∗ if 140 6.1. Introduction and auxiliary results F is lower semicontinuous and convex with F ≥ ·, · on X × X ∗ and gra A = {(x, x∗ ) ∈ X × X ∗ | F (x, x∗ ) = x, x∗ }. Let (z, z ∗ ) ∈ X × X ∗ . Then F(z,z ∗ ) : X × X ∗ → ]−∞, +∞] [55, 57, 74] is defined by (for every (x, x∗ ) ∈ X × X ∗ ) x, z ∗ + z, x∗ + z, z ∗ F(z,z ∗ ) (x, x∗ ) = F (z + x, z ∗ + x∗ ) − = F (z + x, z ∗ + x∗ ) − z + x, z ∗ + x∗ + x, x∗ . (6.1) We recall the following basic fact regarding the second dual ball: Fact 6.1.2 (Goldstine) (See [58, Theorem 2.6.26] or [44, Theorem 3.27].) The weak*-closure of BX in X ∗∗ is BX ∗∗ . Fact 6.1.3 (Borwein) (See [20, Theorem 1] or [92, Theorem 3.1.1].) Let f : X → ]−∞, +∞] be a proper lower semicontinuous and convex function. Let ε > 0 and β ≥ 0 (where 1 0 = ∞). Assume that x0 ∈ dom f and x∗0 ∈ ∂ε f (x0 ). There exist xε ∈ X, x∗ε ∈ X ∗ such that xε − x0 + β | xε − x0 , x∗0 | ≤ x∗ε − x∗0 ≤ √ ε(1 + β x∗0 ), √ ε, x∗ε ∈ ∂f (xε ), | xε − x0 , x∗ε | ≤ ε + √ ε . β Fact 6.1.4 (Simons) (See [73, Theorem 17] or [74, Theorem 37.1].) Let A : X ⇒ X ∗ be maximally monotone and of type (D). Then A is of type (FP). 141 6.2. Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type Fact 6.1.5 (Simons / Marques Alves and Svaiter) (See [71, Lemma 15] or [74, Theorem 36.3(a)], and [56, Theorem 4.4].) Let A : X ⇒ X ∗ be maximally monotone, and let F : X × X ∗ → ]−∞, +∞] be a representative of A. Then the following are equivalent. (i) A is type of (D). (ii) A is of type (NI). (iii) For every (x0 , x∗0 ) ∈ X × X ∗ , inf (x,x∗ )∈X×X ∗ 6.2 F(x0 ,x∗0 ) (x, x∗ ) + 1 2 x 2 + 1 2 x∗ 2 = 0. Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type In Theorem 6.2.1 of this section (see also [7]), we provide an affirmative answer to the following question, posed by S. Simons [73, Problem 18, page 406]: Let A : X ⇒ X ∗ be maximally monotone such that A is of type (FP). Is A necessarily of type (D)? In consequence, in Corollary 6.2.2 we record that the three notions in Definition 6.1.1 actually coincide. Simons posed another question in [74, Problem 47.6]: 142 6.2. Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type Let A : dom A → X ∗ be linear and maximally monotone. Assume that A is of type (FP). Is A necessarily of type (NI)? By Fact 6.1.5, [74, Problem 47.6] is a special case of [73, Problem 18, page 406]. Let A : X ⇒ X ∗ be monotone. For convenience, we defined ΦA on X ∗∗ × X ∗ by ΦA : (x∗∗ , x∗ ) → sup (a,a∗ )∈gra A x∗∗ , a∗ + a, x∗ − a, a∗ . Then we have ΦA |X×X ∗ = FA . The next theorem is our first main result in Chapter 6. In conjunction with the corollary that follows, it provides the affirmative answer promised to Simons’s problem posed in [73, Problem 18, page 406]. Theorem 6.2.1 Let A : X ⇒ X ∗ be maximally monotone such that A is of type (FP). Then A is of type (NI). Proof. After translating the graph if necessary, we can and do suppose that ∗∗ × X ∗ . We must show that ∗ (0, 0) ∈ gra A. Let (x∗∗ 0 , x0 ) ∈ X ∗ ∗∗ ∗ ΦA (x∗∗ 0 , x0 ) ≥ x0 , x0 (6.2) and we consider two cases. Case 1 : x∗∗ 0 ∈ X. Then (6.2) follows directly from Fact 5.1.5. 143 6.2. Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type ∗∗ Case 2 : x∗∗ 0 ∈X X. By Fact 6.1.2, there exists a bounded net (xα )α∈I in X that weak* converges to x∗∗ 0 . Thus, we have M = sup xα < +∞ (6.3) α∈I and ∗ xα , x∗0 → x∗∗ 0 , x0 . (6.4) Now we consider two subcases. Subcase 2.1 : There exists α ∈ I, such that (xα , x∗0 ) ∈ gra A. By definition, ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ΦA (x∗∗ 0 , x0 ) ≥ xα , x0 + x0 , x0 − xα , x0 = x0 , x0 . Hence (6.2) holds. Subcase 2.2 : We have (xα , x∗0 ) ∈ / gra A, ∀α ∈ I. (6.5) Set Uε = [0, x∗0 ] + εUX ∗ , (6.6) where ε > 0. Observe that Uε is open and convex. Since (0, 0) ∈ gra A, we have, by the definition of Uε , 0 ∈ ran A ∩ Uε and x∗0 ∈ Uε . In view of (6.5) 144 6.2. Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type and because A is of type (FP), there exists a net (aα,ε , a∗α,ε ) in gra A such that a∗α,ε ∈ Uε and aα,ε , x∗0 + xα , a∗α,ε − aα,ε , a∗α,ε > xα , x∗0 , ∀α ∈ I. (6.7) Now fix α ∈ I. By (6.7), ∗ ∗∗ ∗ ∗ ∗ aα,ε , x∗0 + x∗∗ 0 , aα,ε − aα,ε , aα,ε > x0 − xα , aα,ε + xα , x0 . Hence, ∗ ∗ ∗∗ ∗ ΦA (x∗∗ 0 , x0 ) > x0 − xα , aα,ε + xα , x0 . (6.8) Since a∗α,ε ∈ Uε , there exist tα,ε ∈ [0, 1] and b∗α,ε ∈ UX ∗ (6.9) a∗α,ε = tα,ε x∗0 + εb∗α,ε . (6.10) such that Using (6.8), (6.10), and (6.3), we deduce that ∗ ∗∗ ∗ ∗ ∗ ΦA (x∗∗ 0 , x0 ) > x0 − xα , tα,ε x0 + εbα,ε + xα , x0 ∗ ∗∗ ∗ ∗ = tα,ε x∗∗ 0 − xα , x0 + ε x0 − xα , bα,ε + xα , x0 ∗ ∗∗ ∗ ≥ tα,ε x∗∗ 0 − xα , x0 − ε x0 − xα + xα , x0 145 6.2. Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type ∗ ∗∗ ≥ tα,ε x∗∗ + M ) + xα , x∗0 . 0 − xα , x0 − ε( x0 (6.11) In view of (6.9) and since α ∈ I was chosen arbitrarily, we take the limit in (6.11) and obtain with the help of (6.4) that ∗ ∗∗ ∗ ΦA (x∗∗ + M ) + x∗∗ 0 , x0 ) ≥ −ε( x0 0 , x0 . (6.12) Next, letting ε → 0 in (6.12), we have ∗∗ ∗ ∗ ΦA (x∗∗ 0 , x0 ) ≥ x0 , x0 . (6.13) Therefore, (6.2) holds in all cases. We now obtain the promised corollary: Corollary 6.2.2 Let A : X ⇒ X ∗ be maximally monotone. Then the following are equivalent. (i) A is of type (D). (ii) A is of type (NI). (iii) A is of type (FP). Proof. “(i)⇒(iii)”: Fact 6.1.4. “(iii)⇒(ii)”: Theorem 6.2.1 . “(ii)⇒(i)”: Fact 6.1.5. Remark 6.2.3 Let A : X ⇒ X ∗ be maximally monotone. Corollary 6.2.2 establishes the equivalences of the key types (D), (NI), and (FP), which as 146 6.3. The adjoint of a maximally monotone linear relation noted all hold when X is reflexive or A = ∂f , where f : X → ]−∞, +∞] is convex, lower semicontinuous, and proper (see [26, 72, 74]). Furthermore, these notions are also equivalent to type (ED), see [76]. For a nonlinear operator they also coincide with the uniqueness of maximal extensions to X ∗∗ (see [56]). In [26, p. 454] there is a discussion of this result and of the linear case. Finally, when A is a linear relation, it has recently been established that all these notions coincide with the monotonicity of the adjoint multifunction A∗ (see Section 6.3). 6.3 The adjoint of a maximally monotone linear relation In this section, we provide tools to give an affirmative answer to a question posed by Phelps and Simons. Phelps and Simons posed the following question in [63, Section 9, item 2]: Let A : dom A → X ∗ be linear and maximally monotone. Assume that A∗ is monotone. Is A necessarily of type (D)? Theorem 6.3.1 Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then A is of type (NI) if and only if A∗ is monotone. Proof. 147 6.3. The adjoint of a maximally monotone linear relation ∗ ∗ “⇒”: Suppose to the contrary that there exists (a∗∗ 0 , a0 ) ∈ gra A such ∗ that a∗∗ 0 , a0 < 0. Then we have sup (a,a∗ )∈gra A ∗ ∗ a, −a∗0 + a∗∗ 0 , a − a, a = sup {− a, a∗ } (a,a∗ )∈gra A ∗ = 0 < −a∗∗ 0 , a0 , which contradicts that A is of type (NI). Hence A∗ is monotone. “⇐”: Define F : X × X ∗ → ]−∞, +∞] : (x, x∗ ) → ιgra A (x, x∗ ) + x, x∗ . Since A is maximally monotone, Fact 3.2.8 implies that F is proper lower semicontinuous and convex, and a representative of A. Let (v0 , v0∗ ) ∈ X×X ∗ . Recalling (6.1), note that F(v0 ,v0∗ ) : (x, x∗ ) → ιgra A (v0 + x, v0∗ + x∗ ) + x, x∗ (6.14) is proper lower semicontinuous and convex. By Fact 5.1.1, there exists (y ∗∗ , y ∗ ) ∈ X ∗∗ × X ∗ such that K := inf (x,x∗ )∈X×X ∗ F(v0 ,v0∗ ) (x, x∗ ) + 2 + 1 2 ∗ ∗ ∗∗ = −F(v )− ∗ (y , y 0 ,v ) 1 2 y ∗∗ = − F(v0 ,v0∗ ) + 0 1 2 · · 1 2 2 ∗ 2 − x 2 + 1 2 x∗ 2 (0, 0) 1 2 y∗ 2 . (6.15) 148 6.3. The adjoint of a maximally monotone linear relation Since (x, x∗ ) → F(v0 ,v0∗ ) (x, x∗ )+ 21 x 2 + 21 x∗ 2 is coercive, there exist M > 0 and a sequence (an , a∗n )n∈N in X × X ∗ such that, ∀n ∈ N, an + a∗n ≤ M (6.16) and F(v0 ,v0∗ ) (an , a∗n ) + <K+ 1 n2 1 2 an 1 2 y∗ + a∗n 1 2 2 ∗ ∗ ∗∗ = −F(v )− ∗ (y , y 0 ,v ) 0 ⇒ F(v0 ,v0∗ ) (an , a∗n ) + + 2 2 < 1 2 2 an + 1 2 y ∗∗ 1 2 a∗n 2 2 − 1 2 y∗ 2 + 1 n2 ∗ ∗ ∗∗ + F(v )+ ∗ (y , y 0 ,v ) 0 (by (6.15) ) 1 2 y ∗∗ 1 n2 (6.17) ∗ ∗ ∗∗ ⇒ F(v0 ,v0∗ ) (an , a∗n ) + F(v ) + an , −y ∗ + a∗n , −y ∗∗ < ∗ (y , y 0 ,v ) 0 ⇒ (y ∗ , y ∗∗ ) ∈ ∂ Set β = 2 ∗ 1 F(v0 ,v0∗ ) (an , an ) n2 1 max{ y ∗ , y ∗∗ }+1 . 1 n2 (by [92, Theorem 2.4.2(ii)]). (6.18) (6.19) Then by Fact 6.1.3, there exist sequences (an , a∗n )n∈N in X × X ∗ and (yn∗ , yn∗∗ )n∈N in X ∗ × X ∗∗ such that, ∀n ∈ N, an − an + a∗n − a∗n + β an − an , y ∗ + a∗n − a∗n , y ∗∗ max{ yn∗ − y ∗ , yn∗∗ − y ∗∗ } ≤ 2 n an − an , yn∗ + a∗n − a∗n , yn∗∗ ≤ ≤ 1 n (6.20) (6.21) 1 n2 + (yn∗ , yn∗∗ ) ∈ ∂F(v0 ,v0∗ ) (an , a∗n ). 1 nβ (6.22) (6.23) Then, ∀n ∈ N, we have an , yn∗ + a∗n , yn∗∗ − an , y ∗ − a∗n , y ∗∗ 149 6.3. The adjoint of a maximally monotone linear relation = an − an , yn∗ + an , yn∗ − y ∗ + a∗n − a∗n , yn∗∗ + a∗n , yn∗∗ − y ∗∗ ≤ an − an , yn∗ + a∗n − a∗n , yn∗∗ + | an , yn∗ − y ∗ | + | a∗n , yn∗∗ − y ∗∗ | ≤ 1 n2 + 1 nβ + an · yn∗ − y ∗ + a∗n · yn∗∗ − y ∗∗ ≤ 1 n2 + 1 nβ + ( an + a∗n ) · max{ yn∗ − y ∗ , yn∗∗ − y ∗∗ } ≤ 1 n2 + 1 nβ + n2 M (by (6.22)) (by (6.16) and (6.21)). (6.24) By (6.20), ∀n ∈ N, we have an − an a∗n − a∗n + ≤ n1 . (6.25) Thus by (6.16), ∀n ∈ N, we have an − an 2 1 n 2 an + a∗n + an − an = ≤ 2 2 − a∗n an + an 1 n + 1 n + 2 a∗n + ≤ n1 (2M + n2 ) = n2 M + 2 a∗n + a∗n a∗n − a∗n 1 n (by (6.25)) 2 . n2 (6.26) Similarly, by (6.21), for all n ∈ N, we have yn∗ yn∗∗ 2 − y∗ 2 2 − y ∗∗ ≤ 2 y∗ + 4 n ≤ 4 n 4 n2 ≤ 4 nβ + 4 , n2 y ∗∗ + 4 n2 ≤ 4 nβ + 4 . n2 1 2 yn∗ (6.27) Thus, ∀n ∈ N, ∗ ∗ ∗∗ F(v0 ,v0∗ ) (an , a∗n ) + F(v ∗ (yn , yn ) + 0 ,v ) 0 1 2 an 2 + 1 2 a∗n 2 + 2 + 1 2 yn∗∗ 2 150 6.3. The adjoint of a maximally monotone linear relation ∗ ∗ ∗∗ = F(v0 ,v0∗ ) (an , a∗n ) + F(v ∗ (yn , yn ) + 0 ,v ) 1 2 0 1 2 + yn∗∗ 1 2 y∗ 1 2 + 1 2 a∗n 2 + 1 2 yn∗ 2 1 2 an 2 1 2 an 2 + 1 2 a∗n 2 + 1 2 a∗n 2 ∗ ∗ ∗∗ + F(v )+ ∗ (y , y 0 ,v ) 1 2 y ∗∗ 2 ∗ ∗ ∗∗ + F(v )+ ∗ (y , y 0 ,v ) 1 2 y ∗∗ 2 0 2 + F(v0 ,v0∗ ) (an , a∗n ) + + 2 2 − F(v0 ,v0∗ ) (an , a∗n ) + + an y∗ 0 2 ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ < F(v0 ,v0∗ ) (an , a∗n ) + F(v ) ∗ (yn , yn ) − F(v0 ,v ∗ ) (an , an ) − F(v ,v ∗ ) (y , y 0 ,v ) 0 0 0 0 + 1 2 an 2 + a∗n 2 − an + 1 2 yn∗ 2 + yn∗∗ 2 − y ∗∗ 2 − a∗n 2 2 − y∗ 2 + 1 n2 (by (6.17)) an , yn∗ + a∗n , yn∗∗ − an , y ∗ − a∗n , y ∗∗ ≤ + 1 2 an 2 − an + 1 2 yn∗ 2 − y∗ 2 2 + a∗n 2 − a∗n + yn∗∗ 2 − y ∗∗ ≤ 1 n2 + 1 nβ + n2 M + n1 M + = 7 n2 + 5 nβ + n3 M. 1 n2 + 4 nβ + 4 n2 (by (6.23)) 2 2 + + 1 n2 1 n2 (by (6.24), (6.26) and (6.27)) (6.28) By (6.23), (6.14), and [92, Theorem 3.2.4(vi)&(ii)], there exists a sequence (zn∗ , zn∗∗ )n∈N in (gra A)⊥ and such that (yn∗ , yn∗∗ ) = (a∗n , an ) + (zn∗ , zn∗∗ ), ∀n ∈ N. (6.29) 151 6.3. The adjoint of a maximally monotone linear relation Since A∗ is monotone and (zn∗∗ , zn∗ ) ∈ gra(−A∗ ), it follows from (6.29) that, ∀n ∈ N, yn∗ , yn∗∗ − yn∗ , an − yn∗∗ , a∗n + a∗n , an = yn∗ − a∗n , yn∗∗ − an (6.30) = zn∗ , zn∗∗ ≤ 0 ⇒ yn∗ , yn∗∗ ≤ yn∗ , an + yn∗∗ , a∗n − a∗n , an . Then by (6.14) and (6.23), we have a∗n , an = F(v0 ,v0∗ ) (an , a∗n ) and, ∀n ∈ N, ∗ ∗ ∗∗ yn∗ , yn∗∗ ≤ yn∗ , an + yn∗∗ , a∗n − F(v0 ,v0∗ ) (an , a∗n ) = F(v ∗ (yn , yn ). (6.31) 0 ,v ) 0 By (6.28) and (6.31), ∀n ∈ N, we have F(v0 ,v0∗ ) (an , a∗n ) + yn∗ , yn∗∗ + < 7 n2 + 2 + 1 2 a∗n 2 a∗n 2 < 7 n2 + inf F(v0 ,v0∗ ) (x, x∗ ) + 1 2 x 2 + 1 2 x∗ 2 ≤ 0. (6.33) inf F(v0 ,v0∗ ) (x, x∗ ) + 1 2 x 2 + 1 2 x∗ 2 ≥ 0. (6.34) 5 nβ 1 2 an + 1 2 + yn∗ 1 2 2 + 1 2 yn∗∗ 2 + n3 M ⇒ F(v0 ,v0∗ ) (an , a∗n ) + 1 2 an 2 5 nβ + n3 M. (6.32) Thus by (6.32), (x,x∗ )∈X×X ∗ By (6.14), (x,x∗ )∈X×X ∗ 152 6.3. The adjoint of a maximally monotone linear relation Combining (6.33) with (6.34), we obtain inf (x,x∗ )∈X×X ∗ F(v0 ,v0∗ ) (x, x∗ ) + 1 2 x 2 + 1 2 x∗ 2 = 0. (6.35) Thus by Fact 6.1.5, A is of type (NI). Remark 6.3.2 The proof of the necessary part of Theorem 6.3.1 follows closely that of [30, Theorem 2]. The proof of the sufficient part of Theorem 6.3.1 was partially inspired by that of [93, Theorem 32.L] and that of [54, Theorem 2.1]. Combining Corollary 6.2.2 and Theorem 6.3.1, we get the following result. Corollary 6.3.3 Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then the following are equivalent. (i) A is of type (D). (ii) A is of type (NI). (iii) A is of type (FP). (iv) A∗ is monotone. Remark 6.3.4 When A is linear and continuous, Corollary 6.3.3 is due to Bauschke and Borwein [4, Theorem 4.1]. Phelps and Simons in [63, Theorem 6.7] considered the case when A is linear but possibly discontinuous; they arrived at some of the implications of Corollary 6.3.3 in that case. 153 6.3. The adjoint of a maximally monotone linear relation Corollary 6.3.3(iv)⇒(i) gives an affirmative answer to a problem posed by Phelps and Simons in [63, Section 9, item 2] on the converse of [63, Theorem 6.7(c)⇒(f )]. It is interesting to compare Corollary 6.3.3 with the following related result by Brezis and Browder. Suppose that X is reflexive and let A : X ⇒ X ∗ be a monotone linear relation with closed graph. Then A is maximally monotone if and only if A∗ is (maximally) monotone; see [28–30] and also the recent works [70, 89]. We conclude with an application of Corollary 6.3.3 to an operator studied previously by Phelps and Simons [63]. Example 6.3.5 Suppose that X = L1 [0, 1] so that X ∗ = L∞ [0, 1], let D = x ∈ X | x is absolutely continuous, x(0) = 0, x ∈ X ∗ , and set A : X ⇒ X∗ : x → {x }, ∅, if x ∈ D; otherwise. By [63, Example 4.3], A is an at most single-valued maximally monotone linear relation with proper dense domain, and A is neither symmetric nor skew. Moreover, dom A∗ = {z ∈ X ∗∗ | z is absolutely continuous, z(1) = 0, z ∈ X ∗ } ⊆ X A∗ z = −z , ∀z ∈ dom A∗ , and A∗ is monotone. Therefore, Corollary 6.3.3 154 6.4. Discussion implies that A is of type (D), of type (NI), and of type (FP). 6.4 Discussion Our first main result (Theorem 6.2.1) in this chapter is obtained by applying Goldstine’s Theorem (see Fact 6.1.2). Simons, Marques Alves and Svaiter’s characterization of type (D) operators and Borwein’s generalization of the Brøndsted-Rockafellar theorem are the main tools for obtaining the other main result (Theorem 6.3.1). Corollary 6.3.3 motivates the following question: Let A : X ⇒ X ∗ be a monotone linear relation with closed graph. Assume that A∗ is monotone. Is A necessarily of type (D)? 155 Chapter 7 Properties of monotone operators and the partial inf convolution of Fitzpatrick functions Chapter 7 is mainly based on the work in [15, 17] by Bauschke, Wang and Yao. Let F1 , F2 : X × X ∗ → ]−∞, +∞]. Then the partial inf-convolution on the second variable F1 F1 2 F2 : 2 F2 , is the function defined on X × X ∗ by (x, x∗ ) → ∗inf ∗ F1 (x, x∗ − y ∗ ) + F2 (x, y ∗ ). y ∈X In this chapter, we study the properties of FA 2 FB for two maximally mono- tone operators A and B. We also consider the connection between FA 2 FB and FA+B . Then we provide a new proof of the following result due to Voisei [83]: Let A, B : X ⇒ X ∗ be maximally monotone linear relations, and suppose that [dom A − dom B] is closed. Then A + B is maximally monotone. 156 7.1. Auxiliary results 7.1 Auxiliary results The next result was first established in [5, Proposition 2.2(v)] by Bauschke, Borwein and Wang in a Hilbert space. Now we generalize it to a general Banach space. Proposition 7.1.1 Let A : X → X ∗ be linear and monotone. Then ∗ ∗ FA (x, x∗ ) = 2qA ( 1 x∗ + 12 A∗ x) = 21 qA (x∗ +A∗ x), + 2 + ∀(x, x∗ ) ∈ X×X. (7.1) ∗ = ran A . If ran A+ is closed, then dom qA + + Proof. By Proposition 3.1.3(ix), dom A∗ ∩ X = X. Hence for every (x, x∗ ) ∈ X × X ∗, FA (x, x∗ ) = sup [ x, Ay + y, x∗ − y, Ay ] y∈X = 2 sup y∈X y, 12 x∗ + 12 A∗ x − qA+ (y) ∗ = 2qA ( 1 x∗ + 12 A∗ x) + 2 ∗ (x∗ + A∗ x). = 21 qA + (7.2) By [92, Proposition 2.4.4(iv) and Theorem 2.3.3], ∗ ran ∂qA+ ⊆ dom ∂qA . + (7.3) By Proposition 3.2.10, ran ∂qA+ = ran A+ . Then by (7.3), ∗ ∗ ran A+ ⊆ dom ∂qA ⊆ dom qA + + (7.4) 157 7.1. Auxiliary results Then by the Brøndsted-Rockafellar Theorem (see [92, Theorem 3.1.2]), ∗ ∗ ⊆ dom qA ⊆ ran A+ . ran A+ ⊆ dom ∂qA + + ∗ By the assumption that ran A+ is closed, we have ran A+ = dom ∂qA = + ∗ . dom qA + Now we give a direct proof of the following result. Fact 7.1.2 (Bartz-Bauschke-Borwein-Reich-Wang) (See [3, Corollary 5.9].) Let C be a closed convex nonempty set of X. Then FNC = ιC ⊕ ι∗C . Proof. Let (x, x∗ ) ∈ X × X ∗ . Then we have FNC (x, x∗ ) = sup (c,c∗ )∈gra N = C [ x, c∗ + c, x∗ − c, c∗ ] C ,k≥0 [ x, kc∗ + c, x∗ − c, kc∗ ] C ,k≥0 [k( x, c∗ − c, c∗ ) + c, x∗ ] sup (c,c∗ )∈gra N = sup (c,c∗ )∈gra N (7.5) By (7.5), (x, x∗ ) ∈ dom FNC ⇒ sup (c,c∗ )∈gra N C [ x, c∗ − c, c∗ ] ≤ 0 inf [− x, c∗ + c, c∗ ] ≥ 0 inf [ c − x, c∗ − 0 ] ≥ 0 ⇔ (c,c∗ )∈gra NC ⇔ (c,c∗ )∈gra NC ⇔ (x, 0) ∈ gra NC (by Fact 5.1.2) 158 7.1. Auxiliary results ⇔ x ∈ C. (7.6) Now assume x ∈ C. By (7.5), FNC (x, x∗ ) = ι∗C (x∗ ). (7.7) Combine (7.6) and (7.7), FNC = ιC ⊕ ι∗C . Following Penot [64], if F : X × X ∗ → ]−∞, +∞], we set F : X ∗ × X : (x∗ , x) → F (x, x∗ ). (7.8) Fact 7.1.3 (Fitzpatrick) (See [45, Proposition 4.2 and Theorem 4.3].) Let A : X ⇒ X ∗ be a monotone operator. Then FA∗ = ·, · on gra A and x ∈ X | ∃x∗ ∈ X ∗ such that FA∗ (x∗ , x) = x, x∗ ⊆ conv(dom A). Fact 7.1.4 (See [92, Theorem 2.4.14].) Let f : X → ]−∞, +∞] be a sublinear function. Then the following hold. (i) ∂f (x) = {x∗ ∈ ∂f (0) | x∗ , x = f (x)}, ∀x ∈ dom f . (ii) If f is lower semicontinuous, then f = sup ·, ∂f (0) . Fact 7.1.5 (Simons and Z˘ alinescu) (See [78, Theorem 4.2].) Let Y be a Banach space and F1 , F2 : X × Y → ]−∞, +∞] be proper, lower semicontinuous, and convex. Assume that for every (x, y) ∈ X × Y , (F1 2 F2 )(x, y) > −∞ 159 7.1. Auxiliary results and that λ>0 λ [PX dom F1 − PX dom F2 ] is a closed subspace of X. Then for every (x∗ , y ∗ ) ∈ X × X ∗ , (F1 ∗ ∗ ∗ 2 F2 ) (x , y ) = min [F1∗ (x∗ − w∗ , y ∗ ) + F2∗ (w∗ , y ∗ )] . ∗ ∗ w ∈X The following result was first established in [21, Theorem 7.4]. Now we give a new proof. Fact 7.1.6 (Borwein) Let A, B : X ⇒ X ∗ be linear relations such that gra A and gra B are closed. Assume that dom A − dom B is closed. Then (A + B)∗ = A∗ + B ∗ . Proof. We have ιgra(A+B) = ιgra A 2 ιgra B . (7.9) Let (x∗∗ , x∗ ) ∈ X ∗∗ × X ∗ . Since gra A and gra B are closed convex, ιgra A and ιgra B are proper lower semicontinuous and convex. Then by Fact 7.1.5 and (7.9), there exists y ∗ ∈ X ∗ such that ιgra(A+B)∗ (x∗∗ , x∗ ) = ι gra(A+B) ⊥ (−x∗ , x∗∗ ) = ι∗gra(A+B) (−x∗ , x∗∗ ) (since gra(A + B) is a subspace) = ι∗gra A (y ∗ , x∗∗ ) + ι∗gra B (−x∗ − y ∗ , x∗∗ ) = ι(gra A)⊥ (y ∗ , x∗∗ ) + ι(gra B)⊥ (−x∗ − y ∗ , x∗∗ ) = ιgra A∗ (x∗∗ , −y ∗ ) + ιgra B ∗ (x∗∗ , x∗ + y ∗ ) 160 7.1. Auxiliary results = ιgra(A∗ +B ∗ ) (x∗∗ , x∗ ). (7.10) Thus gra(A + B)∗ = gra(A∗ + B ∗ ) and hence (A + B)∗ = A∗ + B ∗ . Lemma 7.1.7 Let A, B : X ⇒ X ∗ be maximally monotone, and suppose that λ>0 λ [dom A − dom B] is a closed subspace of X. Set E = x ∈ X | ∃x∗ ∈ X ∗ such that FA∗ (x∗ , x) = x, x∗ and F = x ∈ X | ∃x∗ ∈ X ∗ such that FB∗ (x∗ , x) = x, x∗ . Then λ>0 λ [dom A − dom B] = λ>0 λ [E − F ] . Moreover, if A and B are of type (FPV), then we have λ>0 λ [dom A − dom B] = λ>0 λ [PX dom FA − PX dom FB ] . Proof. Using Fact 7.1.3, we see that λ>0 ⊆ ⊆ λ [dom A − dom B] ⊆ λ>0 λ>0 λ>0 λ [E − F ] λ conv(dom A) − conv(dom B) λ conv(dom A) − conv(dom B) 161 7.1. Auxiliary results = λ>0 ⊆ λ>0 = λ>0 Hence λ[conv(dom A − dom B)] λ [conv(dom A − dom B)] λ [dom A − dom B] λ>0 λ [dom A − dom B] = (using the assumption). λ>0 λ [E − F]. Now assume that A, B are of type (FPV). Then by Fact 5.1.6 and Fact 5.1.13, we have λ>0 ⊆ ⊆ λ [dom A − dom B] ⊆ λ>0 λ>0 = λ>0 λ>0 λ [PX dom FA − PX dom FB ] λ dom A − dom B λ dom A − dom B ⊆ λ [dom A − dom B] λ>0 λ [dom A − dom B] (using the assumption). Corollary 7.1.8 Let A, B : X ⇒ X ∗ be maximally monotone linear relations, and suppose that [dom A − dom B] is a closed subspace. Then λ>0 λ [PX dom FA − PX dom FB ] = [dom A − dom B] = λ>0 λ PX dom FA∗ − PX dom FB∗ . Proof. Apply directly Fact 5.1.14 and Lemma 7.1.7. 162 7.1. Auxiliary results Corollary 7.1.9 Let A : X ⇒ X ∗ be maximally monotone linear relations and C ⊆ X be a closed convex set. Assume that λ>0 λ [dom A − C] is a closed subspace. Then λ>0 λ [PX dom FA − PX dom FNC ] = λ>0 = λ>0 λ [dom A − C] λ PX dom FA∗ − PX dom FN∗ C . Proof. Let B = NC . Then apply directly Fact 5.1.14, Fact 5.1.12 and Lemma 7.1.7. Fact 7.1.10 (See [74, Lemma 23.9] or [10, Proposition 4.2].) Let A, B : X ⇒ X ∗ be monotone operators and dom A∩dom B = ∅. Then FA+B ≤ FA 2 FB . Proof. Let (x, x∗ ) ∈ X × X ∗ and y ∗ ∈ X ∗ . Then we have FA (x, y ∗ ) + FB (x, x∗ − y ∗ ) = + sup (b,b∗ )∈gra B = sup (a,a∗ )∈gra A [ a, y ∗ + x, a∗ − a, a∗ ] [ b, x∗ − y ∗ + x, b∗ − b, b∗ ] sup (a,a∗ )∈gra A,(b,b∗ )∈gra B a, y ∗ + x, a∗ − a, a∗ + b, x∗ − y ∗ + x, b∗ − b, b∗ ≥ sup (a,a∗ )∈gra A,(a,b∗ )∈gra B a, y ∗ + x, a∗ − a, a∗ + a, x∗ − y ∗ + x, b∗ − a, b∗ = sup (a,a∗ )∈gra A,(a,b∗ )∈gra B = FA+B (x, x∗ ). [ a, x∗ + x, a∗ + b∗ − a, a∗ + b∗ ] (7.11) 163 7.1. Auxiliary results Then inf y∗ ∈X ∗ [FA (x, y ∗ ) + FB (x, x∗ − y ∗ )] ≥ FA+B (x, x∗ ) and thus FA 2 FB (x, x ∗) ≥ FA+B (x, x∗ ). We now discover more properties of FA 2 FB . Proposition 7.1.11 was first established by Bauschke, Wang and Yao in [15, Proposition 5.9] when X is a reflexive space. We now provide a nonreflexive version. Proposition 7.1.11 Let A, B : X ⇒ X ∗ be maximally monotone and suppose that λ>0 λ [dom A − dom B] is a closed subspace of X. Then FA 2 FB is proper, norm×weak∗ lower semicontinuous and convex, and the partial infimal convolution is exact everywhere. Proof. Define F1 , F2 : X × X ∗ → ]−∞, +∞] by F1 : (x, x∗ ) → FA∗ (x∗ , x), F2 : (x, x∗ ) → FB∗ (x∗ , x). Since FA , FB is norm-weak∗ lower semicontinuous, F1∗ (x∗ , x) = FA (x, x∗ ), F2∗ (x∗ , x) = FB (x, x∗ ), ∀(x, x∗ ) ∈ X × X ∗ . (7.12) Take (x, x∗ ) ∈ X × X ∗ . By Fact 5.1.5, F1 2 F2 (x, x∗ ) ≥ x, x∗ > −∞. 164 7.1. Auxiliary results In view of Lemma 7.1.7, λ>0 λ [PX dom F1 − PX dom F2 ] = λ>0 λ [dom A − dom B] is a closed subspace. By Fact 7.1.5 and (7.12), F1 2 F2 ∗ (x∗ , x) = min [F1∗ (x∗ − y ∗ , x) + F2∗ (y ∗ , x)] ∗ ∗ y ∈X = min [FA (x, x∗ − y ∗ ) + FB (x, y ∗ )] = FA ∗ ∗ y ∈X Hence FA 2 FB 2 FB (x, x∗ ). is proper, norm×weak∗ lower semicontinuous and convex, and the partial infimal convolution is exact. Proposition 7.1.12 (See [15, Proposition 5.5].) Let X be reflexive and A : X ⇒ X ∗ be a monotone linear relation with nonempty closed graph. Then FA∗ : (x∗ , x) → ιgra A (x, x∗ ) + x, x∗ . Proof. Define g : X × X ∗ → ]−∞, +∞] : (x, x∗ ) → x, x∗ + ιgra A (x, x∗ ). Thus by Fact 3.2.8 and the assumption, g is proper, lower semicontinuous and convex. By definition of FA , FA (x, x∗ ) = g∗ (x∗ , x) (for every (x, x∗ ) ∈ X × X ∗ ). Therefore, by [92, Theorem 2.3.3] we have FA∗ = g. The next new result provides a sufficient but not necessary condition for the maximality of the sum of two maximally monotone operators. Proposition 7.1.13 Let A, B : X ⇒ X ∗ be maximally monotone and suppose that λ>0 λ [PX sume that FA 2 FB dom FA − PX dom FB ] is a closed subspace of X. As- = FA+B . Then A + B is maximally monotone. 165 7.1. Auxiliary results Proof. We first show FA+B ≥ ·, · . (7.13) Let (x, x∗ ) ∈ X × X ∗ and y ∗ ∈ X ∗ . Then by Fact 5.1.5, we have FA (x, y ∗ ) + FB (x, x∗ − y ∗ ) ≥ x, y ∗ + x, x∗ − y ∗ = x, x∗ . Then FA 2 FB (x, x ∗ ) = ∗inf ∗ [FA (x, y ∗ ) + FB (x, x∗ − y ∗ )] ≥ x, x∗ . y ∈X By (7.14) and the assumption that FA 2 FB (7.14) = FA+B , we have (7.13) holds. Combining (7.13) and Fact 5.1.7, A + B is maximally monotone. Let A, B : X ⇒ X ∗ be maximally monotone such that dom A ∩ dom B = ∅. By Fact 7.1.10 FA 2 FB ≥ FA+B . It naturally raises a question: Does the equality always hold under the Rockafellar’s constraint qualification: dom A ∩ int dom B = ∅ (which was also asked by the referee of [90])? The equality has a far-reaching meaning. If this were true, then Proposition 7.1.13 would directly solve the sum problem in the affirmative. However, in general, it cannot hold. The easiest example probably is [10, Example 4.7] by Bauschke, McLaren and Sendov on two projection operators in one dimensional space. Now we give another counterexample on a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous sublinear function, which thus implies that we cannot approach the maximality of the sum of a linear relation A and the subdif166 7.1. Auxiliary results ferential of a proper lower semicontinuous sublinear function f by showing FA 2 F∂f = FA+∂f . Example 7.1.14 Let X be a Hilbert space, BX be the closed unit ball of X and Id be the identity mapping from X to X. Let f : x ∈ X → x . Then we have F∂f 2 FId (x, x ∗ )= x + 0, 1 x + x∗ 4 We also have F∂f +Id = F∂f 2 FId if x + x∗ ≤ 1; 2 − 1 2 x + x∗ + 14 , if x + x∗ > 1. (7.15) when X = R. Proof. By [10, Example 3.10 and Example 3.3], we have FId (x, x∗ ) = 1 4 x + x∗ 2 (7.16) F∂f (x, x∗ ) = x + ιBX (x∗ ), ∀(x, x∗ ) ∈ X × X. (7.17) Note that ∂f (x) = NBX (x) = BX , { x }, x 0, if x = 0; (7.18) otherwise. [0, ∞[ · x, ∅, if x < 1; if x = 1; (7.19) otherwise. 167 7.1. Auxiliary results Indeed, clearly ∂f (0) = BX . Assume x = 0. By Fact 7.1.4(i), x∗ ∈ ∂f (x) ⇔ x∗ ∈ BX , x∗ , x = x ⇔ x∗ = 1, x∗ , x = x · x∗ ⇔ x∗ = x x . Hence (7.18) holds. Similarly, (7.19) holds. Then by (7.16) and (7.17), (F∂f 2 FId )(x, x = x + ∗ ) = inf ∗ y x + x∗ 1 4 2 x + ιBX (y ∗ ) + + 12 inf ∗ y 1 4 x + x∗ − y ∗ x + x∗ , y ∗ + ιBX (y ∗ ) + 1 2 2 y∗ 2 . (7.20) We consider two cases: Case 1 : x + x∗ ≤ 1. Then we directly obtain that x + x∗ , y ∗ + ιBX (y ∗ ) + inf ∗ y And thus, F∂f 2 FId (x, x ∗) 1 2 y∗ 2 = − 21 x + x∗ 2 . = x . Case 2 : x+x∗ > 1. Since K : y ∗ ∈ X → x+x∗ , y ∗ +ιBX (y ∗ )+ 12 y ∗ 2 is convex, y0∗ is a minimizer of K if and only if 0 ∈ x + x∗ + y0∗ + NBX (y0∗ ). Since x + x∗ > 1, by (7.19), y0∗ = 1. Thus by (7.19) again, there exists ρ > 0 such that 0 = x + x∗ + y0∗ + ρy0∗ . Then we have ρ + 1 = x + x∗ and y0∗ = − F∂f 2 FId (x, x ∗) x+x∗ x+x∗ = x + 1 4 . Thus inf K = K(y0∗ ) = − x + x∗ + 12 . Then x + x∗ 2 − 1 2 x + x∗ + 41 . Hence (7.15) holds. 168 7.1. Auxiliary results In order to show F∂f +Id = F∂f 2 FId , we consider the case when X = R. Now we consider the point (−1, 4). Then by (7.15), (F∂f 2 FId )(−1, 4) = 1 + 1 = 2. (7.21) On the other hand, F∂f +Id (−1, 4) = sup [ x, 4 + −1, x + ∂f (x) − x, ∂f (x) + x ] x∈R = sup [ x, 3 + −1, ∂f (x) − x, ∂f (x) + x ] x∈R = sup x, 3 + −1, ∂f (x) − |x| − |x|2 (by Fact 7.1.4(i)) x∈R = max sup x, 3 + −1, ∂f (x) − |x| − |x|2 , x>0 sup x, 3 + −1, ∂f (x) − |x| − |x|2 , x=0 sup x, 3 + −1, ∂f (x) − |x| − |x|2 x<0 = max sup x, 3 − 1 − |x| − |x|2 , 1, sup x, 3 + 1 − |x| − |x|2 x>0 x<0 (by (7.18)) = max sup x, 3 − 1 − x − |x|2 , 1, 1 x>0 = max sup 2x − 1 − |x|2 , 1 x>0 = max{0, 1} = 1 = 2 = F∂f Hence F∂f +Id = F∂f 2 FId (−1, 4) (by (7.21)). 2 FId . 169 7.2. Fitzpatrick function of the sum of two linear relations 7.2 Fitzpatrick function of the sum of two linear relations Section 7.2 is mainly based on the work in [15, 17] by Bauschke, Wang and Yao. Theorem 7.2.1 was first proved in [15, Theorem 5.10] by Bauschke, Wang and Yao in a reflexive space. Now we generalize it to a general Banach space. Theorem 7.2.1 (Fitzpatrick function of the sum) Let A, B : X ⇒ X ∗ be maximally monotone linear relations, and suppose that [dom A − dom B] is closed. Then FA+B = FA 2 FB . Proof. Let (z, z ∗ ) ∈ X × X ∗ . By Fact 7.1.10, it suffices to show that FA+B (z, z ∗ ) ≥ (FA 2 FB )(z, z ∗ ). (7.22) If (z, z ∗ ) ∈ / dom FA+B , then (7.22) clearly holds. Now assume that (z, z ∗ ) ∈ dom FA+B . Then FA+B (z, z ∗ ) = sup [ x, z ∗ + z, x∗ − x, x∗ + z − x, y ∗ − ιgra A (x, x∗ ) {x,x∗ ,y ∗ } − ιgra B (x, y ∗ )]. (7.23) Let Y = X ∗ and define F, K : X × X ∗ × Y → ]−∞, +∞] respectively by F :(x, x∗ , y ∗ ) ∈ X × X ∗ × Y → x, x∗ + ιgra A (x, x∗ ) 170 7.2. Fitzpatrick function of the sum of two linear relations K :(x, x∗ , y ∗ ) ∈ X × X ∗ × Y → x, y ∗ + ιgra B (x, y ∗ ) Then by (7.23), FA+B (z, z ∗ ) = (F + K)∗ (z ∗ , z, z). (7.24) By Fact 3.2.8 and the assumptions, F and K are proper lower semicontinuous and convex, and dom F − dom K = [dom A − dom B] × X ∗ × Y is a closed subspace. Thus by Fact 3.1.2 and (7.24), there exist (z0∗ , z0∗∗ , z1∗∗ ) ∈ X ∗ × X ∗∗ × Y ∗ such that FA+B (z, z ∗ ) = F ∗ (z ∗ − z0∗ , z − z0∗∗ , z − z1∗∗ ) + K ∗ (z0∗ , z0∗∗ , z1∗∗ ) = F ∗ (z ∗ − z0∗ , z, 0) + K ∗ (z0∗ , 0, z) (by (z, z ∗ ) ∈ dom FA+B ) = FA (z, z ∗ − z0∗ ) + FB (z, z0∗ ) ≥ (FA 2 FB )(z, z ∗ ). Thus (7.22) holds and hence FA+B = FA 2 FB . The following result was first established by Voisei in [83]. Simons gave another proof in [74, Theorem 46.3]. Now we give a new approach for showing this result. Theorem 7.2.2 Let A, B : X ⇒ X ∗ be maximally monotone linear rela171 7.2. Fitzpatrick function of the sum of two linear relations tions, and suppose that [dom A − dom B] is closed. Then A+B is maximally monotone. Proof. Combining Theorem 7.2.1, Corollary 7.1.8, and Proposition 7.1.13, we have A + B is maximally monotone. The following examples show that the constraint on the domain in Theorem 7.2.1 cannot be weakened. The rest of this section is all based on the work in [17] by Bauschke, Wang and Yao. Let S be defined in Example 3.3.1, i.e., 2 S : dom S → (N) : y → 1 2 yn + yi , (7.25) n∈N i<n with dom S = y = (yn ) ∈ 2 yi yi = 0, (N) i≥1 i≤n n∈N ∈ 2 (N) . We explicitly compute the Fitzpatrick functions FS+S ∗ , FS , FS ∗ , and show that FS+S ∗ = FS 2 FS ∗ even though S, S ∗ are linear maximally mono- tone with dom S − dom S ∗ being a dense linear subspace in 2 (N). Lemma 7.2.3 Let X be a reflexive space and S : dom S → X ∗ be a maximally monotone skew linear operator. Then FS = ιgra(−S ∗ ) and FS∗∗ = FS ∗ = ιgra S ∗ + ·, · . 172 7.2. Fitzpatrick function of the sum of two linear relations Proof. By Proposition 7.1.12, FS∗ = (ιgra S ) . Then FS = FS∗ ∗ = ιgra S ∗ = ιgra S ∗ = ιgra S −1 ∗ = ι(gra S −1 )⊥ = ιgra(−S ∗ ) . (7.26) From Fact 3.2.12, gra(−S) ⊆ gra S ∗ , we have FS ∗ ≥ F(−S) = ιgra −(−S)∗ = ιgra S ∗ , this shows that dom FS ∗ ⊆ gra S ∗ . By the Brezis-Browder theorem (Fact 3.2.13) and Fact 5.1.5, FS ∗ (x, x∗ ) = x, x∗ ∀(x, x∗ ) ∈ gra S ∗ . Hence FS ∗ = ιgra S ∗ + ·, · . Again by Proposition 7.1.12, FS∗∗ = ιgra S ∗ + ·, · . Theorem 7.2.4 Let H = 2 (N) and S be defined as in Example 3.3.1. Then FS+S ∗ (x, x∗ ) = ιH×{0} (x, x∗ ) 1 s2 , if (x, x∗ ) ∈ dom S ∗ × {0} with s = 2 ∗ FS 2 FS ∗ (x, x ) = ∞ otherwise. Consequently, FS 2 FS ∗ i≥1 xi ; (7.27) = FS+S ∗ . 173 7.2. Fitzpatrick function of the sum of two linear relations Proof. By Fact 3.2.12, (S + S ∗ )|dom S = 0. (7.28) Let (x, x∗ ) ∈ H × H. Using (7.28) and Fact 3.2.12, we have FS+S ∗ (x, x∗ ) = sup a∈dom S x∗ , a = ι(dom S)⊥ (x∗ ) = ι{0} (x∗ ) = ιH×{0} (x, x∗ ). (7.29) Then by Fact 7.1.10, we have (FS 2 FS ∗ )(x, x ∗ ) = ∞, x∗ = 0. (7.30) It follows from Lemma 7.2.3 that (FS 2 FS ∗ )(x, 0) = inf {FS (x, y ∗ ) + FS ∗ (x, −y ∗ )} ∗ y ∈H = inf {ιgra(−S ∗ ) (x, y ∗ ) + ιgra S ∗ (x, −y ∗ ) + x, −y ∗ } ∗ y ∈H = inf {ιgra S ∗ (x, −y ∗ ) + x, −y ∗ }. ∗ y ∈H Thus, FS s= 2 FS ∗ (x, 0) i≥1 xi . (7.31) = ∞ if x ∈ / dom S ∗ . Now suppose x ∈ dom S ∗ and Then by (7.31) and Proposition 3.3.6, we have FS 2 FS ∗ (x, 0) = x, S ∗ x = 12 s2 . Combine the results above, (7.27) holds. Since dom S ∗ = H, FS 2 FS ∗ = FS+S ∗ . Let A : H ⇒ H be a maximally monotone linear relation. Then [15, 174 7.2. Fitzpatrick function of the sum of two linear relations Theorem 7.6] shows that: A∗ = −A if and only if dom A = dom A∗ and FA = FA∗ . Let A = S ∗ with S defined as in Example 3.3.1. Lemma 7.2.3 shows that FA = FA∗ , but A∗ = S = −S ∗ = −A by Proposition 3.3.5. Hence the requirement dom A = dom A∗ cannot be omitted. Let V be the Volterra integral operator. In the rest of this section, we systematically study T = V −1 and its adjoint T ∗ . We compute the Fitzpatrick functions FT , FT ∗ , FT +T ∗ , and we show that FT 2 FT ∗ = FT +T ∗ . This shows that the constraint qualification for the formula of the Fitzpatrick function of the sum of two maximally monotone operators cannot be significantly weakened either. To study Fitzpatrick functions of sums of maximally monotone operators, we need: Lemma 7.2.5 Let H = L2 [0, 1] and V be the Volterra integration operator defined in Example 3.4.4 and e ≡ 1 ∈ L2 [0, 1]. Then qV∗ + (z) = ιspan{e} (z) + z, e 2 , ∀z ∈ L2 [0, 1]. Proof. Let z ∈ H. By Example 3.4.4(iv) and Fact 7.1.1, we have qV∗ + (z) = ∞, if z ∈ / span{e}. Now suppose that z = le for some l ∈ R. By Example 3.4.4(iv), qV∗ + (z) = sup { x, z − qV+ (x)} = sup { x, le − x∈H = l2 = le, e x∈H 2 1 4 x, e 2 } = z, e 2 . 175 7.2. Fitzpatrick function of the sum of two linear relations Hence qV∗ + (z) = ιspan{e} (z) + z, e 2 . Lemma 7.2.6 Let H = L2 [0, 1] and T be defined as in Theorem 3.4.5. We have (for every (x, y ∗ ) ∈ H × H) FT (x, y ∗ ) = FV (y ∗ , x) = ιspan{e} (x + V ∗ y ∗ ) + 1 2 x + V ∗y∗, e 2 , FT ∗ (x, y ∗ ) = FV ∗ (y ∗ , x) = ιspan{e} (x + V y ∗ ) + 1 2 x + V y∗, e 2 . (7.32) Proof. Apply Fact 5.1.5, Fact 7.1.1 and Lemma 7.2.5 to obtain the formula for FT . Let (x, y ∗ ) ∈ H × H. By Proposition 3.1.3(iv), Fact 7.1.1 and Lemma 7.2.5 again, we have FT ∗ (x, y ∗ ) = FV ∗ (y ∗ , x) = 12 qV∗ +∗ (x + V ∗∗ y ∗ ) = 21 qV∗ + (x + V y ∗ ) = ιspan{e} (x + V y ∗ ) + 1 2 x + V y∗, e 2 . Remark 7.2.7 Theorem 7.2.8 below gives another example showing that FT +T ∗ = FT 2 FT ∗ while T, T ∗ are maximally monotone, and dom T − dom T ∗ is a dense subspace in L2 [0, 1]. This again shows that the assumption that dom A − dom B is closed in Theorem 7.2.1 cannot be weakened substantially. Theorem 7.2.8 Let H = L2 [0, 1] and T be defined as in Theorem 3.4.5, e ≡ 1 ∈ L2 [0, 1] and set C = {x ∈ L2 [0, 1] : x is absolutely continuous, and x ∈ L2 [0, 1]}. 176 7.2. Fitzpatrick function of the sum of two linear relations Then (FT FT +T ∗ (x, x∗ ) = ιH×{0} (x, x∗ ), ∀(x, x∗ ) ∈ H × H 1 x(1)2 + x(0)2 , if (x, x∗ ) ∈ C × {0}; 2 ∗ ∗ F )(x, x ) = 2 T ∞, otherwise. Consequently, FT 2 FT ∗ (7.33) = FT +T ∗ . Proof. By Theorem 3.4.5(i) and Example 3.4.4(iii), (T + T ∗ )y = 0, ∀y ∈ dom T ∩ dom T ∗ = {V x | x ∈ e⊥ }. (7.34) Let (x, x∗ ) ∈ H × H. Using Theorem 3.4.5(iii) and (7.34), we see that FT +T ∗ (x, x∗ ) = sup y∈dom T ∩dom T ∗ x∗ , y = sup x∗ , y y∈H = ι{0} (x∗ ) = ιH×{0} (x, x∗ ). (7.35) By Fact 7.1.10, we have FT 2 FT ∗ (x, x∗ ) = ∞, ∀ x∗ = 0. (7.36) When x∗ = 0, by Lemma 7.2.6, FT 2 FT ∗ (x, 0) = inf {FT (x, y ∗ ) + FT ∗ (x, −y ∗ )} ∗ y ∈H = inf {ιspan{e} (x + V ∗ y ∗ ) + ∗ y ∈H + ιspan{e} (x − V y ∗ ) + 1 2 1 2 x + V ∗y∗, e (7.37) 2 x − V y ∗ , e 2 }. 177 7.3. Fitzpatrick function of the sum of a linear relations and a normal cone operator Observe that x + V ∗ y ∗ ∈ span{e}, x − V y ∗ ∈ span{e} ⇔ x − V y ∗ + V y ∗ + V ∗ y ∗ ∈ span{e}, x − V y ∗ ∈ span{e} ⇔ x − V y ∗ ∈ span{e}, (by Example 3.4.4(iv)) ⇔ x ∈ V y ∗ + span{e} ⇔ x is absolutely continuous and y ∗ = x . Therefore, (FT 2 FT ∗ )(x, 0) = ∞ if x ∈ / C. For x ∈ C, using (7.37) and the fact that x − V x = x(0)e and x + V ∗ x = x(1)e, we obtain FT 2 FT ∗ (x, 0) = 1 2 = 12 x(1)2 + 21 x(0)2 = x + V ∗x , e 1 2 + 1 2 x − V x ,e 2 x(1)2 + x(0)2 . Thus, (7.33) holds. Consequently, FT 7.3 2 2 FT ∗ = FT +T ∗ . Fitzpatrick function of the sum of a linear relations and a normal cone operator The proof of Theorem 7.3.1 partially follows that of [16, Theorem 3.1] by Bauschke, Wang and Yao. Theorem 7.3.1 Let A : X ⇒ X ∗ be a maximally monotone linear relation, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Then FA+NC = FA 2 FNC . 178 7.3. Fitzpatrick function of the sum of a linear relations and a normal cone operator Proof. Let (z, z ∗ ) ∈ X × X ∗ . By Fact 7.1.10, it suffices to show that FA+NC (z, z ∗ ) ≥ (FA 2 FNC )(z, z ∗ ). (7.38) By Corollary 5.3.4, PX [dom FA+NC ] ⊆ [dom(A + NC )] ⊆ C. Thus, (7.38) holds if z ∈ / C. Now assume that z ∈ C. Set g : X × X ∗ → ]−∞, +∞] : (x, x∗ ) → x, x∗ + ιgra A (x, x∗ ). (7.39) By Fact 3.2.8, g is convex. Hence, h = g + ιC×X ∗ (7.40) c0 ∈ dom A ∩ int C, (7.41) is convex as well. Let and let c∗0 ∈ Ac0 . Then (c0 , c∗0 ) ∈ gra A∩(int C×X ∗ ) = dom g∩int dom ιC×X ∗ . By Fact 5.1.4, ιC×X ∗ is continuous at (c0 , c∗0 ). Then, FA+NC (z, z ∗ ) = sup (x,x∗ ,c∗ ) x, z ∗ + z, x∗ − x, x∗ + z − x, c∗ − ιgra A (x, x∗ ) − ιgra NC (x, c∗ ) ≥ sup [ x, z ∗ + z, x∗ − x, x∗ − ιgra A (x, x∗ ) − ιC×X ∗ (x, x∗ )] (x,x∗ ) 179 7.4. Discussion = sup [ x, z ∗ + z, x∗ − h(x, x∗ )] (x,x∗ ) = h∗ (z ∗ , z) = g∗ (y ∗ , y ∗∗ ) + ι∗C×X ∗ (z ∗ − y ∗ , z − y ∗∗ ) (by Fact 5.1.1, ∃(y ∗ , y ∗∗ ) ∈ X ∗ × X ∗∗ ) = g∗ (y ∗ , y ∗∗ ) + ι∗C (z ∗ − y ∗ ) + ι{0} (z − y ∗∗ ). We consider two cases: Case 1 : z = y ∗∗ . Clearly, FA+NC (z, z ∗ ) = +∞ ≥ (FA 2 FNC )(z, z ∗ ). Case 2 : z = y ∗∗ . Then FA+NC (z, z ∗ ) ≥ g ∗ (y ∗ , y ∗∗ ) + ι∗C (z ∗ − y ∗ ) = FA (z, y ∗ ) + ι∗C (z ∗ − y ∗ ) = FA (z, y ∗ ) + ι∗C (z ∗ − y ∗ ) + ιC (z) ≥ FA = (FA 2 FNC )(z, z ∗ + ι∗C ) ) (by Fact 7.1.2). Hence (7.38) holds and thus FA+NC = FA 7.4 2 (ιC 2 FNC . Discussion It would be interesting to find out whether Theorem 7.3.1 generalizes to the following: Let A : X ⇒ X ∗ be a maximally monotone linear relation, let C be a nonempty closed convex subset of X. Assume that dom A − λ>0 λC is a closed subspace of X. Is it necessarily true that FA+NC = FA 2 FNC ? 180 Chapter 8 BC–functions and examples of type (D) operators This chapter is based on the work in [8] by Bauschke, Borwein, Wang and Yao. We first introduce some notation related to this chapter. Let F : X × X ∗ → ]−∞, +∞]. We say F is a BC–function (BC stands for “bigger conjugate”) [74] if F is proper and convex with F ∗ (x∗ , x) ≥ F (x, x∗ ) ≥ x, x∗ ∀(x, x∗ ) ∈ X × X ∗ . (8.1) Let Y be a real Banach space, and let F1 , F2 : X × Y → ]−∞, +∞]. Then the function F1 F1 1 F2 is defined on X × Y by 1 F2 : (x, y) → inf {F1 (u, y) + F2 (x − u, y)}. u∈X In Example 8.3.1(iii)&(v) of this chapter, we provide a negative answer to the following question posed by S. Simons [74, Problem 22.12]: Let F1 , F2 : X × X ∗ → ]−∞, +∞] be lower semicontinuous BC– 181 8.1. Auxiliary results functions and λ>0 Is F1 8.1 λ [PX ∗ dom F1 − PX ∗ dom F2 ] is a closed subspace of X ∗ . 1 F2 necessarily a BC–function? Auxiliary results Fact 8.1.1 (Banach and Mazur) (See [44, Theorem 5.17]).) Every separable Banach space is isometric to a subspace of C[0, 1]. Fact 8.1.2 (Fitzpatrick) (See [45, Corollary 3.9 and Proposition 4.2].) Let A : X ⇒ X ∗ be maximally monotone. Then FA is a BC–function and FA = ·, · on gra A. Let Y be a real Banach space. Let L : X → Y be linear. We say L is an isomorphism into Y if L is one to one, continuous and L−1 is continuous on ran L. We say L is an isometry if Lx = x , ∀x ∈ X. The spaces X, Y are called isometric if there exists an isometry from X onto Y . Let A : X ⇒ X ∗ be monotone and S be a subspace of X. We say A is S–saturated if Ax + S ⊥ = Ax, ∀x ∈ dom A. Fact 8.1.3 (Simons and Z˘ alinescu) (See [74, Theorem 16.4(b)].) Let Y be a Banach space and F1 , F2 : X × Y → ]−∞, +∞] be proper, 182 8.1. Auxiliary results lower semicontinuous and convex. Assume that for every (x, y) ∈ X × Y , (F1 and that λ>0 λ [PY 1 F2 )(x, y) > −∞ dom F1 − PY dom F2 ] is a closed subspace of Y . Then for every (x∗ , y ∗ ) ∈ X ∗ × Y ∗ , (F1 ∗ ∗ ∗ 1 F2 ) (x , y ) = min [F1∗ (x∗ , u∗ ) + F2∗ (x∗ , y ∗ − u∗ )] . ∗ ∗ u ∈Y Fact 8.1.4 (Simons) (See [74, Theorem 28.9].) Let Y be a real Banach space, and L : Y → X be continuous and linear with ran L closed and ran L∗ = Y ∗ . Let A : X ⇒ X ∗ be monotone with dom A ⊆ ran L such that gra A = ∅. Then A is maximally monotone if and only if A is ran L– saturated and L∗ AL is maximally monotone. Fact 8.1.5 (See [58, Theorem 3.1.22(b)] or [44, Exercise 2.39(i), page 59].) Let Y be a real Banach space. Assume that L : Y → X is an isomorphism into X. Then ran L∗ = Y ∗ . Corollary 8.1.6 Let Y be a real Banach space, and L : Y → X be an isomorphism into X. Let T : Y ⇒ Y ∗ be monotone. Then T is maximally monotone if, and only if (L∗ )−1 T L−1 is maximally monotone. Proof. Let A = (L∗ )−1 T L−1 . Then dom A ⊆ ran L. Since L is an isomorphism into X, ran L is closed. By Fact 8.1.5, ran L∗ = Y ∗ . Hence gra(L∗ )−1 T L−1 = ∅ if and only if gra T = ∅. Clearly, A is monotone. Since (0, (ran L)⊥ ) ⊆ gra(L∗ )−1 , A = (L∗ )−1 T L−1 is ran L–saturated. By 183 8.1. Auxiliary results Fact 8.1.4, A = (L∗ )−1 T L−1 is maximally monotone if and only if L∗ AL = T is maximally monotone. The following result will allow us for constructing operators that are not of type (D) in different Banach spaces. Corollary 8.1.7 Let Y be a real Banach space, and L : Y → X be an isomorphism into X. Let T : Y ⇒ Y ∗ be maximally monotone. Assume that T is not of type (D). Then (L∗ )−1 T L−1 is maximally monotone but is not of type (D). Proof. By Corollary 8.1.6, (L∗ )−1 T L−1 is maximally monotone. By Fact 6.1.5 or Corollary 6.2.2 , there exists (y0∗∗ , y0∗ ) ∈ Y ∗∗ × Y ∗ such that sup (b,b∗ )∈gra T y0∗∗ , b∗ + y0∗ , b − b, b∗ < y0∗∗ , y0∗ . By Fact 8.1.5, there exists x∗0 ∈ X ∗ such that L∗ x∗0 = y0∗ . (8.2) Let A = (L∗ )−1 T L−1 . Then we have sup (a,a∗ )∈gra A = L∗∗ y0∗∗ , a∗ + x∗0 , a − a, a∗ sup (Ly,a∗ )∈gra A = sup (Ly,a∗ )∈gra A = sup (Ly,a∗ )∈gra A = sup (y,y ∗ )∈gra T y0∗∗ , L∗ a∗ + x∗0 , Ly − Ly, a∗ y0∗∗ , L∗ a∗ + L∗ x∗0 , y − y, L∗ a∗ y0∗∗ , L∗ a∗ + y0∗ , y − y, L∗ a∗ y0∗∗ , y ∗ + y0∗ , y − y, y ∗ (by (Ly, a∗ ) ∈ gra A ⇔ (y, L∗ a∗ ) ∈ gra T ) 184 8.2. Main construction < y0∗∗ , y0∗ (by (8.2)) = L∗∗ y0∗∗ , x∗0 . (8.3) Thus A is not type (NI) and hence A = (L∗ )−1 T L−1 is not type (D) by Fact 6.1.5. 8.2 Main construction We shall give an abstract framework for constructing non type (D) operators in non-reflexive spaces. Lemma 8.2.1 Let A : X ⇒ X ∗ be a skew linear relation. Then FA = ιgra(−A∗ )∩X×X ∗ . (8.4) Proof. Let (x0 , x∗0 ) ∈ X × X ∗ . We have FA (x0 , x∗0 ) = = sup (x,x∗ )∈gra A sup (x,x∗ )∈gra A { (x∗0 , x0 ), (x, x∗ ) − x, x∗ } (x∗0 , x0 ), (x, x∗ ) = ι(gra A)⊥ (x∗0 , x0 ) = ιgra(−A∗ ) (x0 , x∗0 ) = ιgra(−A∗ )∩X×X ∗ (x0 , x∗0 ). Hence (8.4) holds. 185 8.2. Main construction The main result in this chapter is Theorem 8.2.2, which our constructed examples are based on. Theorem 8.2.2 Let A : X ∗ → X ∗∗ be linear and continuous. Assume that ran A ⊆ X and there exists e ∈ X ∗∗ \X such that Ax∗ , x∗ = e, x∗ 2 , ∀x∗ ∈ X ∗ . Let P and S respectively be the symmetric part and antisymmetric part of A. Let T : X ⇒ X ∗ be defined by gra T = {(−Sx∗ , x∗ ) | x∗ ∈ X ∗ , e, x∗ = 0} = {(−Ax∗ , x∗ ) | x∗ ∈ X ∗ , e, x∗ = 0}. (8.5) Let f : X → ]−∞, +∞] be a proper lower semicontinuous and convex function. Set F = f ⊕ f ∗ on X × X ∗ . Then the following hold. (i) T is maximally monotone. (ii) gra T ∗ = {(Sx∗ + re, x∗ ) | x∗ ∈ X ∗ , r ∈ R}. (iii) T is not of type (D). (iv) FT = ιC , where C = {(−Ax∗ , x∗ ) | x∗ ∈ X ∗ }. (8.6) (v) If dom T ∩ int dom ∂f = ∅, then T + ∂f is maximally monotone. 186 8.2. Main construction (vi) F and FT are BC–functions on X × X ∗ . (vii) Moreover, λ>0 λ PX ∗ (dom FT ) − PX ∗ (dom F ) = X ∗ . Assume that there exists (v0 , v0∗ ) ∈ X × X ∗ such that f ∗ (v0∗ ) + f ∗∗ (v0 − A∗ v0∗ ) < v0 , v0∗ . Then FT 1F (8.7) is not a BC–function. (viii) Assume that ran A − λ>0 λ dom f that ∅ = dom f ∗∗ ◦ A∗ |X ∗ is a closed subspace of X and {e}⊥ . Then T + ∂f is not of type (D). Proof. (i): Now we claim that P x∗ = x∗ , e e, ∀x∗ ∈ X ∗ . (8.8) Since ·, e e = ∂( 12 ·, e 2 ) and by [63, Theorem 5.1], ·, e e is a symmetric operator on X ∗ . Clearly, A − ·, e e is skew. Then (8.8) holds. Let x∗ ∈ X ∗ with e, x∗ = 0. Then we have Sx∗ = x∗ , e e + Sx∗ = P x∗ + Sx∗ = Ax∗ ∈ ran A ⊆ X. Thus (8.5) holds and T is well defined. 187 8.2. Main construction We have S is skew and hence T is skew. Let (z, z ∗ ) ∈ X × X ∗ be monotonically related to gra T . By Fact 3.2.9, we have 0 = z, x∗ + −Sx∗ , z ∗ = z + Sz ∗ , x∗ , ∀x∗ ∈ {e}⊥ . Thus by Fact 3.1.1, we have z + Sz ∗ ∈ ({e}⊥ )⊥ = span{e} and then z = −Sz ∗ + κe, ∃κ ∈ R. (8.9) κ z ∗ , e = −Sz ∗ + κe, z ∗ = z, z ∗ ≥ 0. (8.10) By Fact 3.2.9 again, Then by (8.9) and (8.8), Az ∗ = P z ∗ + Sz ∗ = P z ∗ + κe − z = [ z ∗ , e + κ] e − z. (8.11) By the assumptions that z ∈ X, Az ∗ ∈ X and e ∈ / X, [ z ∗ , e + κ] = 0 by (8.11). Then by (8.10), we have z ∗ , e = κ = 0 and thus (z, z ∗ ) ∈ gra T by (8.9). Hence T is maximally monotone. ∗∗ × X ∗ . Then we have ∗ (ii): Let (x∗∗ 0 , x0 ) ∈ X ∗ ∗ ∗ ∗ ∗ ∗∗ (x∗∗ = 0, 0 , x0 ) ∈ gra T ⇔ x0 , Sx + x , x0 ∗ ⇔ x∗ , x∗∗ 0 − Sx0 = 0, ∀x∗ ∈ {e}⊥ ∗ ⊥ ⇔ x∗∗ 0 − Sx0 ∈ ({e}⊥ ) = span{e} ∗ ⇔ x∗∗ 0 − Sx0 = re, ∀x∗ ∈ {e}⊥ (by Fact 3.1.1) ∃r ∈ R. 188 8.2. Main construction Thus gra T ∗ = {(Sx∗ + re, x∗ ) | x∗ ∈ X ∗ , r ∈ R}. (iii): By (ii), T ∗ is not monotone. Then by Corollary 6.3.3, T is not of type (D). (iv): By (ii), we have (z, z ∗ ) ∈ gra(−T ∗ ) ∩ X × X ∗ ⇔ (z, z ∗ ) = (−Sz ∗ − re, z ∗ ), z ∈ X, ∃r ∈ R, z ∗ ∈ X ∗ ⇔ (z, z ∗ ) = (−Sz ∗ − z ∗ , e e + [ z ∗ , e − r] e, z ∗ ), ⇔ (z, z ∗ ) = (−Az ∗ + [ z ∗ , e − r] e, z ∗ ), ⇔ (z, z ∗ ) = (−Az ∗ , z ∗ ), ∃r ∈ R, z ∗ ∈ X ∗ ∃r ∈ R, z ∗ ∈ X ∗ (by (8.8)) z∗, e = r (by z, Az ∗ ∈ X and e ∈ / X), ∃r ∈ R, z ∗ ∈ X ∗ ⇔ (z, z ∗ ) ∈ {(−Ax∗ , x∗ ) | x∗ ∈ X ∗ } = C. Thus by Lemma 8.2.1, we have FT = ιC . (v): Apply (i) and Theorem 5.3.1. (vi): Clearly, F is a BC–function. By (i) and Fact 8.1.2, we have FT is a BC–function. (vii): By (iv), we have λ>0 λ PX ∗ (dom FT ) − PX ∗ (dom F ) = X ∗ . (8.12) Then for every (x, x∗ ) ∈ X × X ∗ and u ∈ X, by (vi), FT (x − u, x∗ ) + F (u, x∗ ) = FT (x − u, x∗ ) + (f ⊕ f ∗ )(u, x∗ ) 189 8.2. Main construction ≥ x − u, x∗ + u, x∗ = x, x∗ . Hence (FT 1 F )(x, x ∗ ) ≥ x, x∗ > −∞. (8.13) Then by (8.12), (8.13) and Fact 8.1.3, (FT ∗ ∗ 1 F ) (v0 , v0 ) = min FT∗ (v0∗ , x∗∗ ) + F ∗ (v0∗ , v0 − x∗∗ ) x∗∗ ∈X ∗∗ ≤ FT∗ (v0∗ , A∗ v0∗ ) + F ∗ (v0∗ , v0 − A∗ v0∗ ) = 0 + F ∗ (v0∗ , v0 − A∗ v0∗ ) (by (iv)) = (f ⊕ f ∗ )∗ (v0∗ , v0 − A∗ v0∗ ) = (f ∗ ⊕ f ∗∗ )(v0∗ , v0 − A∗ v0∗ ) = f ∗ (v0∗ ) + f ∗∗ (v0 − A∗ v0∗ ) < v0∗ , v0 (by 8.7). (8.14) Hence FA 1F is not a BC–function. (viii): By the assumption, there exists x∗0 ∈ dom f ∗∗ ◦ A∗ |X ∗ such that e, x∗0 = 0. Let ε0 = e,x∗0 2 2 . By [92, Theorem 2.4.4(iii)]), there exists y0∗∗∗ ∈ ∂ε0 f ∗∗ (A∗ x∗0 ). By [92, Theorem 2.4.2(ii)]), f ∗∗ (A∗ x∗0 ) + f ∗∗∗ (y0∗∗∗ ) ≤ A∗ x∗0 , y0∗∗∗ + ε0 . (8.15) 190 8.2. Main construction Then by [74, Lemma 45.15] or the proof of [67, Eq.(2.5) in Proposition 1], there exists y0∗ ∈ X ∗ such that f ∗∗ (A∗ x∗0 ) + f ∗ (y0∗ ) < A∗ x∗0 , y0∗ + 2ε0 . (8.16) Let z0∗ = y0∗ + x∗0 . Then by (8.16), we have f ∗∗ (A∗ x∗0 ) + f ∗ (z0∗ − x∗0 ) < A∗ x∗0 , z0∗ − x∗0 + 2ε0 = A∗ x∗0 , z0∗ − A∗ x∗0 , x∗0 + 2ε0 = A∗ x∗0 , z0∗ − x∗0 , Ax∗0 + 2ε0 = A∗ x∗0 , z0∗ − 2ε0 + 2ε0 = A∗ x∗0 , z0∗ . (8.17) Then for every (x, x∗ ) ∈ X × X ∗ and u∗ ∈ X, by (vi), FT (x, x∗ − u∗ ) + F (x, u∗ ) = FT (x, x∗ − u∗ ) + (f ⊕ f ∗ )(x, u∗ ) ≥ x, x∗ − u∗ + x, u∗ = x, x∗ . Hence (FT 2 F )(x, x ∗ ) ≥ x, x∗ > −∞. (8.18) Then by (8.18), (iv) and Fact 7.1.5, (FT ∗ ∗ ∗ ∗ 2 F ) (z0 , A x0 ) 191 8.3. Examples and applications = min FT∗ (y ∗ , A∗ x∗0 ) + F ∗ (z0∗ − y ∗ , A∗ x∗0 ) ∗ ∗ y ∈X ≤ FT∗ (x∗0 , A∗ x∗0 ) + F ∗ (z0∗ − x∗0 , A∗ x∗0 ) = 0 + F ∗ (z ∗ − x∗0 , A∗ x∗0 ) (by (iv)) = (f ⊕ f ∗ )∗ (z0∗ − x∗0 , A∗ x∗0 ) = f ∗ (z0∗ − x∗0 ) + f ∗∗ (A∗ x∗0 ) < z0∗ , A∗ x∗0 (by (8.17)). (8.19) Let F0 : X × X ∗ → ]−∞, +∞] be defined by (x, x∗ ) → x, x∗ + ιgra(T +∂f ) (x, x∗ ). Clearly, FT 2F ≤ F0 on X × X ∗ and thus (FT ∗ 2F ) (8.20) ≥ F0∗ on X ∗ × X ∗∗ . By (8.19), F0∗ (z0∗ , A∗ x∗0 ) < z0∗ , A∗ x∗0 . Hence T + ∂f is not of type (NI) and thus T + ∂f is not of type (D) by Fact 6.1.5. 8.3 Examples and applications Example 8.3.1 Suppose that X = c0 , with norm and X ∗∗ = ∞ (N) · ∞ so that X ∗ = · with norm lim sup αn = 0, and let Aα : 1 (N) → ∗. i>n (N) with norm Let α = (αn )n∈N ∈ ∞ (N) αn αi x∗i , (Aα x∗ )n = α2n x∗n + 2 1 · 1, ∞ (N) with be defined by ∀x∗ = (x∗n )n∈N ∈ 1 (N). 192 8.3. Examples and applications Let Pα and Sα respectively be the symmetric part and antisymmetric part of Aα . Let Tα : c0 ⇒ X ∗ be defined by gra Tα = (−Sα x∗ , x∗ ) x∗ ∈ X ∗ , α, x∗ = 0 = (−Aα x∗ , x∗ ) x∗ ∈ X ∗ , α, x∗ = 0 = αn αi x∗i + (− i>n αn αi x∗i )n∈N , x∗ i<n x∗ ∈ X ∗ , α, x∗ = 0 . (8.21) Then the following hold. (i) Aα x∗ , x∗ = α, x∗ 2 , ∀x∗ = (x∗n )n∈N ∈ 1 (N). Hence (8.21) is well defined. (ii) Tα is a maximally monotone operator that is not of type (D). (iii) FTα 1( · ⊕ ιBX ∗ ) is not a BC–function. (iv) Tα + ∂ · (v) If √1 2 < α is a maximally monotone operator that is not of type (D). ∗ ≤ 1, then FTα 1 1( 2 · 2 ⊕ 1 2 · 2) 1 is not a BC–function. (vi) Tα + λJ is a maximally monotone operator that is not of type (D) for every λ > 0. (vii) There exists a linear operator L : c0 → C[0, 1] that is an isometry from c0 to a subspace of C[0, 1]. Then for every λ > 0, (L∗ )−1 (Tα +∂ · )L−1 and (L∗ )−1 (Tα + λJ)L−1 are maximally monotone operators that are not of type (D). 193 8.3. Examples and applications (viii) Let G : 1 (N) → G(x∗ ) ∞ (N) n∈N be Gossez’s operator [50] defined by = i>n Then Te : c0 ⇒ 1 (N) x∗i − x∗i , ∀(x∗n )n∈N ∈ i<n 1 (N). as defined by gra Te = {(−G(x∗ ), x∗ ) | x∗ ∈ 1 (N), x∗ , e = 0} is a maximally monotone operator that is not of type (D), where e = (1, 1, . . . , 1, . . .). Proof. We have α ∈ / c0 . Since α = (αn )n∈N ∈ continuous and ran Aα ⊆ c0 ⊆ ∞ (N), Aα is linear and ∞ (N). (i): We have Aα x∗ , x∗ = x∗n (α2n x∗n + 2 n n αn αi x∗n x∗i n=i αn x∗n )2 = α, x∗ 2 , =( αn αi x∗n x∗i n i>n n i>n α2n x∗n 2 + = α2n x∗n 2 + 2 αn αi x∗i ) = n ∀x∗ = (x∗n )n∈N ∈ 1 (N). (8.22) Then the proof of Theorem 8.2.2 shows that the symmetric part Pα of Aα is Pα x∗ = α, x∗ α (for every x∗ ∈ 1 (N)). Thus, the skew part Sα of Aα is (Sα x∗ )n∈N = (Aα x∗ )n∈N − (Pα x∗ )n∈N = α2n x∗n + 2 = i>n αn αi x∗i − αn αi x∗i i<n n∈N . i>n αn αi x∗i − αn αi x∗i i (8.23) 194 n∈N 8.3. Examples and applications Then by Theorem 8.2.2, (8.21) is well defined. (ii): Combine Theorem 8.2.2(i)&(iii). (iii): Let f = · on X = c0 . Then f ∗ = ιBX ∗ by [92, Corol- lary 2.4.16]. Since α = 0, there exists i0 ∈ N such that αi0 = 0. Let ei0 = (0, . . . , 0, 1, 0, . . .), i.e., the i0 th component is 1 and the others are 0. Then by (8.23), we have Sα ei0 = αi0 (α1 , . . . , αi0 −1 , 0, −αi0 +1 , −αi0 +2 , . . .). (8.24) Then A∗α ei0 = Pα ei0 − Sα ei0 = αi0 (0, . . . , 0, αi0 , 2αi0 +1 , 2αi0 +2 , . . .). (8.25) Now set v0∗ = ei0 and v0 = 3 α 2∗ ei0 . Thus by (8.25), v0 − A∗α v0∗ = 3 α 2∗ ei0 − A∗α ei0 = (0, . . . , 0, 3 α 2 ∗ − α2i0 , −2αi0 αi0 +1 , −2αi0 αi0 +2 , . . .) (8.26) We have f ∗ (v0∗ ) + f ∗∗ (v0 − A∗α ei0 ) = ιBX ∗ (ei0 ) + v0 − A∗α ei0 = 3 α ∗ ei0 − A∗α ei0 <3 α 2 ∗ ∗ ∗ (by (8.26)) = v0 , v0∗ . 195 8.3. Examples and applications Hence by Theorem 8.2.2 (vii), FTα · ⊕ ιBX ∗ ) is not a BC–function. on X. Since dom f ∗∗ = X ∗∗ , ∅ = dom f ∗∗ ◦ A∗α |X ∗ · (iv): Let f = 1( {e}⊥ . Then apply Theorem 8.2.2(v)&(viii) directly. (v): Let f = √1 2 < α ∗ 1 2 · 2 on X = c0 . Then f ∗ = 1 2 · 2 1 and f ∗∗ = 1 2 · 2. ∗ By ≤ 1, take |αi0 |2 > 21 . Let ei0 be defined as in the proof of (iii). Then set v1∗ = 12 ei0 and v1 = 1 + 12 α2i0 ei0 . By (8.25), we have v1 − A∗α v1∗ = (0, . . . , 0, 1, −αi0 αi0 +1 , −αi0 αi0 +2 , . . .) Since |αi0 αj | ≤ α 2 ∗ (8.27) ≤ 1, ∀j ∈ N, then v1 − A∗α v1∗ ≤ 1. (8.28) We have f ∗ (v1∗ ) + f ∗∗ (v1 − A∗α v1∗ ) = ≤ < 1 8 + α2i 0 4 1 2 + 1 2 v1∗ 2 1 2 ⊕ + 1 2 v1 − A∗α v1∗ 1 2 · 2) ∗ 2 ∗ (by (8.28)) 1 2 1 (by α2i0 > ) 2 = v1∗ , v1 . Hence by Theorem 8.2.2(vii), FTα (vi): Let λ > 0 and f = λ 2 · 2 1 1( 2 · is not a BC–function. on X = c0 . Then f ∗∗ = λ 2 · 2. ∗ The rest of the proof is very similar to that of (iv). 196 8.4. Discussion (vii) : Since c0 is separable by [58, Example 1.12.6] or [44, Proposition 1.26(ii)], by Fact 8.1.1, there exists a linear operator L : c0 → C[0, 1] that is an isometry from c0 to a subspace of C[0, 1]. Then combine (iv), (vi) and Corollary 8.1.7. (viii): Apply (ii) . Remark 8.3.2 The maximal monotonicity of the operator Te in Example 8.3.1(viii) was established by Voisei and Z˘ alinescu in [87, Example 19] and then later a direct proof was given by Bueno and Svaiter in [32, Lemma 2.1]. Bueno and Svaiter also proved that Te is not of type (D) in [32]. Here we give a short and direct proof of the above results. Example 8.3.1(iii)&(v) provide a negative answer to Simons’ problem in [74, Problem 22.12]. 8.4 Discussion The idea of the construction of the operator A in (Theorem 8.2.2) comes from [4, Theorem 5.1] by Bauschke and Borwein. The main tool involved in the main result (Theorem 8.2.2) is Simons and Z˘ alinescu’s version of Attouch-Brezis theorem. 197 Chapter 9 On Borwein-Wiersma decompositions of monotone linear relations This chapter is mainly based on [18] by Bauschke, Wang and Yao, in which although we worked in a reflexive Banach space in [18], we can adapt most results from a reflexive space to a general Banach space. It is well known that every square matrix can be decomposed into the sum of a symmetric matrix and an antisymmetric matrix, where the symmetric part is a gradient of a quadratic function. In this chapter, we provide the necessary and sufficient conditions for a maximally monotone linear relation to be Borwein-Wiersma decomposable, i.e., to be the sum of a subdifferential operator and a skew operator. We also show that Borwein-Wiersma decomposability implies Asplund decomposability. 198 9.1. Decompositions 9.1 Decompositions Definition 9.1.1 (Borwein-Wiersma decomposition [27]) The setvalued operator A : X ⇒ X ∗ is Borwein-Wiersma decomposable if A = ∂f + S, (9.1) where f : X → ]−∞, +∞] is proper lower semicontinuous and convex, and where S : X ⇒ X ∗ is skew and at most single-valued. The right side of (9.1) is a Borwein-Wiersma decomposition of A. Note that every single-valued linear monotone operator A with full domain is Borwein-Wiersma decomposable, with Borwein-Wiersma decomposition A = A+ + A◦ = ∇qA + A◦ . (9.2) Definition 9.1.2 (Asplund irreducibility [1]) The set-valued operator A : X ⇒ X ∗ is irreducible (sometimes termed “acyclic” [27]) if whenever A = ∂f + S, with f : X → ]−∞, +∞] proper lower semicontinuous and convex, and S : X ⇒ X ∗ monotone, then necessarily ran(∂f )|dom A is a singleton. As we shall see in Section 9.1, the following decomposition is less restrictive. Definition 9.1.3 (Asplund decomposition [1]) The set-valued operator 199 9.1. Decompositions A : X ⇒ X ∗ is Asplund decomposable if A = ∂f + S, (9.3) where f : X → ]−∞, +∞] is proper, lower semicontinuous, and convex, and where S is irreducible. The right side of (9.3) is an Asplund decomposition of A. The following fact, due to Censor, Iusem and Zenios [36, 53], was previously known in Rn . Here we give a different proof and extend the result to Banach spaces. Fact 9.1.4 (Censor, Iusem and Zenios) The subdifferential operator of a proper lower semicontinuous convex function f : X → ]−∞, +∞] is paramonotone, i.e., if x∗ ∈ ∂f (x), y ∗ ∈ ∂f (y), (9.4) and x∗ − y ∗ , x − y = 0, (9.5) then x∗ ∈ ∂f (y) and y ∗ ∈ ∂f (x). Proof. By (9.5), x∗ , x + y ∗ , y = x∗ , y + y ∗ , x . (9.6) By (9.4), f ∗ (x∗ ) + f (x) = x∗ , x , f ∗ (y ∗ ) + f (y) = y ∗ , y . 200 9.1. Decompositions Adding them, followed by using (9.6), yields f ∗ (x∗ ) + f (y) + f ∗ (y ∗ ) + f (x) = x∗ , y + y ∗ , x , [f ∗ (x∗ ) + f (y) − x∗ , y ] + [f ∗ (y ∗ ) + f (x) − y ∗ , x ] = 0. Since each bracketed term is nonnegative, we must have f ∗ (x∗ ) + f (y) = x∗ , y and f ∗ (y ∗ ) + f (x) = y ∗ , x . It follows that x∗ ∈ ∂f (y) and that y ∗ ∈ ∂f (x). The following result provides a powerful criterion for determining whether a given operator is irreducible and hence Asplund decomposable. Theorem 9.1.5 Let A : X ⇒ X ∗ be monotone and at most single-valued. Suppose that there exists a dense subset D of dom A such that Ax − Ay, x − y = 0 ∀x, y ∈ D. Then A is irreducible and hence Asplund decomposable. Proof. Let a ∈ D and D := D − {a}. Define A : dom A − {a} → A(· + a). Then A is irreducible if and only if A is irreducible. Now we show A is irreducible. By assumptions, 0 ∈ D and A x − A y, x − y = 0 ∀x, y ∈ D . Let A = ∂f + R, where f is proper lower semicontinuous and convex, and R is monotone. Since A is single-valued on dom A , we have that ∂f and R 201 9.1. Decompositions are single-valued on dom A and that R = A − ∂f on dom A . By taking x∗0 ∈ ∂f (0), rewriting A = (∂f − x∗0 ) + (x∗0 + R), we can and do suppose ∂f (0) = {0}. For x, y ∈ D we have A x − A y, x − y = 0. Then for x, y ∈ D 0 ≤ R(x) − R(y), x − y = A x − A y, x − y − ∂f (x) − ∂f (y), x − y = − ∂f (x) − ∂f (y), x − y . On the other hand, ∂f is monotone, thus, ∂f (x) − ∂f (y), x − y = 0, ∀x, y ∈ D . (9.7) Using ∂f (0) = {0}, ∂f (x) − 0, x − 0 = 0, ∀x ∈ D . (9.8) As ∂f is paramonotone by Fact 9.1.4, ∂f (x) = {0} so that x ∈ argmin f . This implies that D ⊆ argmin f since x ∈ D was chosen arbitrarily. As f is lower semicontinuous, argmin f is closed. Using that D is dense in dom A , it follows that dom A ⊆ D ⊆ argmin f . Since ∂f is single-valued on dom A , ∂f (x) = {0}, ∀x ∈ dom A . Hence we have A is irreducible, and so is A. Remark 9.1.6 In Theorem 9.1.5, the assumption that A be at most single202 9.1. Decompositions valued is important: indeed, let L be a proper subspace of Rn . Then ∂ιL is a linear relation and skew, yet ∂ιL = ∂ιL + 0 is not irreducible. Theorem 9.1.5 and the definitions of the two decomposabilities now yield the following. Corollary 9.1.7 Let A : X ⇒ X ∗ be maximally monotone such that A is Borwein-Wiersma decomposable. Then A is Asplund decomposable. We proceed to give a few sufficient conditions for a maximally monotone linear relation to be Borwein-Wiersma decomposable. The following simple observation will be needed. Lemma 9.1.8 Let A : X ⇒ X ∗ be a monotone linear relation such that A is Borwein-Wiersma decomposable, say A = ∂f +S, where f : X → ]−∞, +∞] is proper, lower semicontinuous, and convex, and where S : X ⇒ X ∗ is at most single-valued and skew. Then the following hold. ∂f (x), if x ∈ dom A; (i) ∂f + Idom A : x → is a monotone linear ∅, otherwise relation. (ii) dom A ⊆ dom ∂f ⊆ dom f ⊆ (A0)⊥ . (iii) If A is maximally monotone, then dom A ⊆ dom ∂f ⊆ dom f ⊆ dom A. (iv) If A is maximally monotone and dom A is closed, then dom ∂f = dom A = dom f . 203 9.1. Decompositions Proof. (i): Indeed, on dom A, we see that ∂f = A − S is the difference of two linear relations. (ii): Clearly dom A ⊆ dom ∂f . As S0 = 0, we have A0 = ∂f (0). Thus, ∀x∗ ∈ A0, x ∈ X, x∗ , x ≤ f (x) − f (0). Then σA0 (x) ≤ f (x) − f (0), where σA0 is the support function of A0. If x ∈ (A0)⊥ , then σA0 (x) = +∞ since A0 is a linear subspace, so f (x) = +∞, ∀x ∈ (A0)⊥ . Therefore, dom f ⊆ (A0)⊥ . Altogether, (ii) holds. (iii): Combine (ii) with Proposition 3.2.2(i). (iv): Apply (iii). Theorem 9.1.9 Let A : X ⇒ X ∗ be a maximally monotone linear relation such that dom A ⊆ dom A∗ . Then A is Borwein-Wiersma decomposable via A = ∂qA + S, where S is an arbitrary linear single-valued selection of A◦ . Moreover, ∂qA = A+ on dom A. Proof. From Proposition 3.2.10(i), A+ is monotone and qA+ = qA , using Proposition 3.2.10(ii), gra A+ ⊆ gra ∂qA+ = gra ∂qA . Let S : dom A → X ∗ be a linear selection of A◦ (the existence of which is guaranteed by a standard Zorn’s lemma argument). Then, S is skew. Thus, by Proposition 3.2.2(v), we have gra A = gra(A+ + S) ⊆ gra(∂qA + S). Since A is maximally monotone, A = ∂qA +S, which is the announced Borwein-Wiersma decomposition. Moreover, ∂qA = A − S = A+ on dom A. 204 9.1. Decompositions Corollary 9.1.10 Let A : X ⇒ X ∗ be a maximally monotone linear relation such that A is symmetric. Then A is Borwein-Wiersma decomposable, with decompositions A = ∂qA + 0. If X is reflexive, then A−1 is Borwein∗ + 0. Wiersma decomposable with A−1 = ∂qA Proof. Using Proposition 3.2.11, we obtain A = A∗ |X . rem 9.1.9 applies; in fact, A = ∂qA . Hence, Theo- If X is reflexive, then we have ∗ by [92, Theorem 2.4.4(iv) and Theorem 2.3.1(iv)]. From A−1 = ∂qA ∗ = ∂qA Proposition 3.1.3(iv), we have A−1 = (A∗ )−1 = (A−1 )∗ . Then A−1 = ∂qA−1 . ∗. Hence A−1 = ∂qA−1 = ∂qA Corollary 9.1.11 Let A : X ⇒ X ∗ be a maximally monotone linear relation such that dom A is closed, and let S be a single-valued linear selection of A◦ . Then qA = qA , A+ = ∂qA is maximally monotone, and A and A∗ |X are Borwein-Wiersma decomposable, with decompositions A = A+ + S and A∗ |X = A+ − S, respectively. Proof. Proposition 3.2.2(iv) implies that dom A∗ |X = dom A. By Proposition 3.2.10(iv), A∗ |X is maximally monotone. In view of Proposition 3.2.2(v), A = A+ + A◦ and A∗ |X = A+ − A◦ . Theorem 9.1.9 yields the BorweinWiersma decomposition A = ∂qA +S. Hence dom A ⊆ dom ∂qA ⊆ dom qA ⊆ dom A = dom A. In turn, since dom A = dom A+ and qA = qA+ , this implies that dom A+ = dom ∂qA+ = dom qA+ . In view of Proposition 3.2.10(i)&(ii), qA+ = qA+ and gra A+ ⊆ gra ∂qA+ . By Theorem 9.1.9, A+ = ∂qA on dom A. Since dom A = dom A+ = dom ∂qA and qA = qA+ = qA+ = qA , this implies that A+ = ∂qA = ∂qA everywhere. Therefore, A+ is maximally monotone. Then we obtain the Borwein-Wiersma decomposition A∗ |X = A+ − S. 205 9.1. Decompositions Theorem 9.1.12 Let A : X ⇒ X ∗ be a maximally monotone linear relation such that A is skew, and let S be a single-valued linear selection of A. Then A is Borwein-Wiersma decomposable via ∂ιdom A + S. Proof. Clearly, S is skew. Proposition 3.1.3(ii) and Proposition 3.2.2(iii) imply that A = A0 + S = (dom A)⊥ + S = ∂ιdom A + S, as announced. Alternatively, by [80, Lemma 2.2], dom A ⊆ dom A∗ and now apply Theorem 9.1.9. Under a mild constraint qualification, the sum of two Borwein-Wiersma decomposable operators is also Borwein-Wiersma decomposable and the decomposition of the sum is the corresponding sum of the decompositions. Proposition 9.1.13 (sum rule) Let A1 and A2 be maximally monotone linear relations from X to X ∗ . Suppose that A1 and A2 are BorweinWiersma decomposable via A1 = ∂f1 + S1 and A2 = ∂f2 + S2 , respectively. Suppose that dom A1 −dom A2 is closed. Then A1 +A2 is Borwein-Wiersma decomposable via A1 + A2 = ∂(f1 + f2 ) + (S1 + S2 ). Proof. By Lemma 9.1.8(iii), dom A1 ⊆ dom f1 ⊆ dom A1 and dom A2 ⊆ dom f2 ⊆ dom A2 . Hence dom A1 − dom A2 ⊆ dom f1 − dom f2 ⊆ dom A1 − dom A2 ⊆ dom A1 − dom A2 = dom A1 − dom A2 . Thus, dom f1 − dom f2 = dom A1 − dom A2 is a closed subspace of X. By [74, Theorem 18.2], ∂f1 + ∂f2 = ∂(f1 + f2 ); furthermore, S1 + S2 is clearly skew. The result thus follows. 206 9.2. Uniqueness results 9.2 Uniqueness results The main result in this section (Theorem 9.2.8) states that if a maximally monotone linear relation A is Borwein-Wiersma decomposable, then the subdifferential part of its decomposition is unique on dom A. We start by showing that subdifferential operators that are monotone linear relations are actually symmetric, which is a variant of a well known result from Calculus. Lemma 9.2.1 Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex. Suppose that the maximally monotone operator ∂f is a linear relation with closed domain. Then ∂f = (∂f )∗ . Proof. Set A = ∂f and Y = dom f . Since dom A is closed, the BrøndstedRockafellar Theorem (see [74, Theorem 18.6]) implies that dom f = Y = dom A. By Proposition 3.2.2(iv), dom A∗ |X = dom A. Let x ∈ Y and consider the directional derivative g = f (x; ·), i.e., g : X → [−∞, +∞] : y → lim t↓0 By [92, Theorem 2.1.14], dom g = r≥0 r f (x + ty) − f (x) . t · (dom f − x) = Y . On the other hand, f is lower semicontinuous on X. Thus, since Y = dom f is a Banach space, f |Y is continuous by [92, Theorem 2.2.20(b)]. Altogether, in view of [92, Theorem 2.4.9], g|Y is continuous. Hence g is lower semicontinuous. Using [92, Corollary 2.4.15] and Fact 3.1.3(v), we now deduce that (∀y ∈ Y ) g(y) = sup ∂f (x), y = Ax, y = x, A∗ y . We thus have proved that (∀x ∈ Y )(∀y ∈ Y ) f (x; y) = Ax, y = x, A∗ y . (9.9) 207 9.2. Uniqueness results In particular, f |Y is differentiable. Now fix x, y, z in Y . Then, using (9.9), we see that Az, y = lim s↓0 A(x + sz), y − Ax, y f (x + sz; y) − f (x; y) = lim s↓0 s s (9.10) = lim lim s↓0 t↓0 f (x + sz + ty) − f (x + sz) f (x + ty) − f (x) . − st st Set h : R → R : s → f (x + sz + ty) − f (x + sz). Since f |Y is differentiable, so is h. For s > 0, the Mean Value Theorem thus yields rs,t ∈ ]0, s[ such that f (x + sz + ty) − f (x + sz) f (x + ty) − f (x) − s s h(s) h(0) = − = h (rs,t ) s s (9.11) = f (x + rs,t z + ty; z) − f (x + rs,t z; z) = t Ay, z . Combining (9.10) with (9.11), we deduce that Az, y = Ay, z . Thus, A is symmetric. The result now follows from Proposition 3.2.11. To improve Lemma 9.2.1, we need the following “shrink and dilate” technique. Lemma 9.2.2 Let A : X ⇒ X ∗ be a monotone linear relation, and let Z be a closed subspace of dom A. Set B = (A + IZ ) + Z ⊥ . Then B is maximally monotone and dom B = Z. 208 9.2. Uniqueness results Proof. Since Z ⊆ dom A and B = A + ∂ιZ it is clear that B is a monotone linear relation with dom B = Z. By Proposition 3.2.2 (i), we have Z ⊥ ⊆ B0 = A0 + Z ⊥ ⊆ (dom A)⊥ + Z ⊥ ⊆ Z ⊥ + Z ⊥ = Z ⊥ . Hence B0 = Z ⊥ = (dom B)⊥ . Therefore, by Proposition 3.2.2(ii), B is maximally monotone. Theorem 9.2.3 Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex, and let Y be a linear subspace of X. Suppose that ∂f + IY is a linear relation. Then ∂f + IY is symmetric. Proof. Put A = ∂f + IY . Assume that (x, x∗ ), (y, y ∗ ) ∈ gra A. Set Z = span{x, y}. Let B : X ⇒ X ∗ be defined as in Lemma 9.2.2. Clearly, gra B ⊆ gra ∂(f +ιZ ). In view of the maximal monotonicity of B, we see that B = ∂(f + ιZ ). Since dom B = Z is closed, it follows from Lemma 9.2.1 that B = B ∗ . In particular, we obtain that x∗ , y = y ∗ , x . Hence, ∂f (x), y = ∂f (y), x and therefore ∂f + IY is symmetric. Lemma 9.2.4 Let A : X ⇒ X ∗ be a maximally monotone linear relation such that A is Borwein-Wiersma decomposable. Then dom A ⊆ dom A∗ . Proof. By hypothesis, there exists a proper lower semicontinuous and convex function f : X → ]−∞, +∞] and an at most single-valued skew operator S such that A = ∂f + S. Hence dom A ⊆ dom S, and Theorem 9.2.3 implies that (A − S) + Idom A is symmetric. Let x and y be in dom A. Then Ax − 2Sx, y = Ax − Sx, y − Sx, y = Ay − Sy, x − Sx, y 209 9.2. Uniqueness results = Ay, x − Sy, x − Sx, y = Ay, x , which implies that (A − 2S)x ⊆ A∗ x. Therefore, dom A = dom(A − 2S) ⊆ dom A∗ . Remark 9.2.5 We can now derive part of the conclusion of Proposition 9.1.13 differently as follows. Since dom A1 −dom A2 is closed, Voisei proved in [83] (see Theorem 7.2.2 or [74, Theorem 46.3]) that A1 + A2 is maximally monotone; moreover, Fact 7.1.6 yields (A1 +A2 )∗ = A∗1 +A∗2 . Using Lemma 9.2.4, we thus obtain dom(A1 + A2 ) = dom A1 ∩ dom A2 ⊆ dom A∗1 ∩ dom A∗2 = dom(A∗1 + A∗2 ) = dom(A1 + A2 )∗ . Therefore, A1 + A2 is Borwein-Wiersma decomposable by Theorem 9.1.9. Theorem 9.2.6 (characterization of subdifferential operators) Let A : X ⇒ X ∗ be a monotone linear relation. Then A is maximally monotone and symmetric ⇔ there exists a proper lower semicontinuous convex function f : X → ]−∞, +∞] such that A = ∂f . Proof. “⇒”: Proposition 3.2.10(ii). “⇐”: Apply Theorem 9.2.3 with Y = X. Remark 9.2.7 Theorem 9.2.6 generalizes [63, Theorem 5.1] of Phelps and Simons. Theorem 9.2.8 (uniqueness of the subdifferential part) Let A : X ⇒ X ∗ be a maximally monotone linear relation such that A is BorweinWiersma decomposable. Then on dom A, the subdifferential part in the de- 210 9.2. Uniqueness results composition is unique and equals to A+ , and the skew part must be a linear selection of A◦ . Proof. Let f1 and f2 be proper lower semicontinuous convex functions from X to ]−∞, +∞], and let S1 and S2 be at most single-valued skew operators from X to X ∗ such that A = ∂f1 + S1 = ∂f2 + S2 . (9.12) Set D = dom A. Since S1 and S2 are single-valued on D, we have A − S1 = ∂f1 and A − S2 = ∂f2 on D. Hence ∂f1 + ID and ∂f2 + ID are monotone linear relations with (∂f1 + ID )(0) = (∂f2 + ID )(0) = A0. (9.13) By Theorem 9.2.3, ∂f1 + ID and ∂f2 + ID are symmetric, i.e., (∀x ∈ D)(∀y ∈ D) ∂f1 (x), y = ∂f1 (y), x and ∂f2 (x), y = ∂f2 (y), x . Thus, (∀x ∈ D)(∀y ∈ D) ∂f2 (x) − ∂f1 (x), y = ∂f2 (y) − ∂f1 (y), x . (9.14) On the other hand, by (9.12), (∀x ∈ D) S1 x − S2 x ∈ ∂f2 (x) − ∂f1 (x). Then by Fact 3.2.2(iii), Proposition 3.2.1(ii) and Proposition 3.1.3(v), (∀x ∈ D)(∀y ∈ D) ∂f2 (x) − ∂f1 (x), y = S1 x − S2 x, y (9.15) 211 9.2. Uniqueness results = − S1 y − S2 y, x = − ∂f2 (y) − ∂f1 (y), x . Now fix x ∈ D. Combining (9.14) and (9.15), we get (∀y ∈ D) ∂f2 (x) − ∂f1 (x), y = 0. Using Fact 3.2.2(iii), we see that ∂f2 (x) − ∂f1 (x) ⊆ D ⊥ = (dom A)⊥ = A0. Hence, in view of Lemma 9.1.8(i), (9.13), and Fact 3.1.3(ii), ∂f1 + ID = ∂f2 + ID . By Lemma 9.2.4 and Theorem 9.1.9, we consider the case when f2 = qA so that ∂f2 = A+ on D. Hence ∂f1 = A+ on D and, if x ∈ D, then S1 x ∈ Ax − ∂f1 (x) = Ax − A+ x = A◦ x by Proposition 3.2.2(v). Remark 9.2.9 In a Borwein-Wiersma decomposition, the skew part need not be unique: indeed, assume that X = R2 , set Y := R × {0}, and let S be given by gra S = (x, 0), (0, x) | x ∈ R . Then S is skew and the maximally monotone linear relation ∂ιY has two distinct Borwein-Wiersma decompositions, namely ∂ιY + 0 and ∂ιY + S. Proposition 9.2.10 Let A : X ⇒ X ∗ be a maximally monotone linear relation. Suppose that A is Borwein-Wiersma decomposable, with subdifferential part ∂f , where f : X → ]−∞, +∞] is proper, lower semicontinuous and convex. Then there exists a constant α ∈ R such that the following hold. 212 9.2. Uniqueness results (i) f = qA + α on dom A. (ii) If dom A is closed, then f = qA + α = qA + α on X. Proof. Let S be a linear single-valued selection of A◦ . By Lemma 9.2.4, dom A ⊆ dom A∗ . In turn, Theorem 9.1.9 yields A = ∂qA + S. Let {x, y} ⊂ dom A. By Theorem 9.2.8, ∂f + Idom A = ∂qA + Idom A . Now set Z = span{x, y}, apply Lemma 9.2.2 to the monotone linear relation ∂f + Idom A = ∂qA + Idom A , and let B be as in Lemma 9.2.2. Note that gra B = gra(∂qA + ∂ιZ ) ⊆ gra ∂(qA + ιZ ) and that gra B = gra(∂f + ∂ιZ ) ⊆ gra ∂(f + ιZ ). By the maximal monotonicity of B, we conclude that B = ∂(qA + ιZ ) = ∂(f + ιZ ). By [67, Theorem B], there exists α ∈ R such that f + ιZ = qA + ιZ + α. Hence α = f (x) − qA (x) = f (y) − qA (y) and repeating this argument with y ∈ (dom A) {x}, we see that f = qA + α on dom A (9.16) and (i) is thus established. Now assume in addition that dom A is closed. Applying Lemma 9.1.8(iv) with both ∂f and ∂qA , we obtain dom qA = dom ∂qA = dom A = dom ∂f = dom f. Consequently, (9.16) now yields f = qA +α. Finally, Corollary 9.1.11 implies that qA = qA . 213 9.3. Characterizations and examples 9.3 Characterizations and examples The following characterization of the Borwein-Wiersma decomposability of a maximally monotone linear relation is quite pleasing. Theorem 9.3.1 (Borwein-Wiersma decomposability) Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then the following are equivalent. (i) A is Borwein-Wiersma decomposable. (ii) dom A ⊆ dom A∗ . (iii) A = A+ + A◦ . Proof. “(i)⇒(ii)”: Lemma 9.2.4. “(i)⇐(ii)”: Theorem 9.1.9. “(ii)⇒(iii)”: Proposition 3.2.2(v). “(ii)⇐(iii)”: This is clear. Corollary 9.3.2 Assume X is reflexive. Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then both A and A∗ are Borwein-Wiersma decomposable if and only if dom A = dom A∗ . Proof. Combine Theorem 9.3.1, Fact 3.2.13, and Fact 3.1.3(vi). We shall now provide two examples of a linear relation S in the Hilbert space to illustrate that the following do occur: • S is Borwein-Wiersma decomposable, but S ∗ is not. • Neither S nor S ∗ is Borwein-Wiersma decomposable. • S is not Borwein-Wiersma decomposable, but S −1 is. 214 9.3. Characterizations and examples 2 (N), Example 9.3.3 Suppose that X is the Hilbert space 1 2 yn S : dom S → X : y → + and set , yi (9.17) n∈N i<n with dom S = y = (yn )n∈N ∈ X yi yi = 0, n∈N i≤n i≥1 ∈X . Then S ∗ : dom S ∗ → X : y → 1 2 yn + yi (9.18) n∈N i>n where dom S ∗ = y = (yn )n∈N ∈ X yi i>n n∈N ∈X . Then S can be identified with an at most single-valued linear relation such that the following hold. (See [63, Theorem 2.5] and Proposition 3.3.2, Proposition 3.3.3, Proposition 3.3.5, and Theorem 3.3.8.) (i) S is maximally monotone and skew. (ii) S ∗ is maximally monotone but not skew. (iii) dom S is dense in 2 (N), and dom S dom S ∗ . (iv) S ∗ = −S on dom S. In view of Theorem 9.3.1, S is Borwein-Wiersma decomposable while S ∗ is not. However, both S and S ∗ are irreducible and Asplund decomposable by 215 9.3. Characterizations and examples Theorem 9.1.5. Because S ∗ is irreducible but not skew, we see that the class of irreducible operators is strictly larger than the class of skew operators. Example 9.3.4 (Inverse Volterra operator) (See Example 3.4.4 and Theorem 3.4.5.) Suppose that X is the Hilbert space L2 [0, 1], and consider the Volterra integration operator (see, e.g., [52, Problem 148]), which is defined by t V : X → X : x → V x, where V x : [0, 1] → R : t → x, (9.19) 0 and set A = V −1 . Then V ∗ : X → X : x → V ∗ x, where V ∗ x : [0, 1] → R : t → 1 x, t and the following hold. (i) We have dom A = x ∈ X x is absolutely continuous, x(0) = 0, and x ∈ X and A : dom A → X : x → x . (ii) We have dom A∗ = x ∈ X x is absolutely continuous, x(1) = 0, 216 9.3. Characterizations and examples and x ∈ X and A∗ : dom A∗ → X : x → −x . (iii) Both A and A∗ are maximally monotone linear operators. (iv) Neither A nor A∗ is symmetric. (v) Neither A nor A∗ is skew. (vi) dom A ⊆ dom A∗ , and dom A∗ ⊆ dom A. (vii) Y = dom A ∩ dom A∗ is dense in X. (viii) Both A + IY and A∗ + IY are skew. By Theorem 9.1.5, both A and A∗ are irreducible and Asplund decomposable. On the other hand, by Theorem 9.3.1, neither A nor A∗ is Borwein-Wiersma decomposable. Finally, A−1 = V and (A∗ )−1 = V ∗ are Borwein-Wiersma decomposable since they are continuous linear operators with full domain. Remark 9.3.5 (an answer to Borwein and Wiersma’s question) The operators S, S ∗ , A, and A∗ defined in this section are all irreducible and Asplund decomposable, but none of them has full domain. This provides an answer to [27, Question (4) in Section 7]: Can one exhibit an irreducible operator whose domain is not the whole space? 217 9.4. When X is a Hilbert space 9.4 When X is a Hilbert space Throughout this short section, we suppose that X is a Hilbert space. Recall (see, e.g., [42, Chapter 5] for basic properties) that if C is a nonempty closed convex subset of X, then the (nearest point) projector PC is well defined and continuous. If Y is a closed subspace of X, then PY is linear and PY = PY∗ . Definition 9.4.1 Let A : X ⇒ X be a maximally monotone linear relation. We define QA by QA : dom A → X : x → PAx x. Note that QA is monotone and a single-valued selection of A because (∀x ∈ dom A) Ax is a nonempty closed convex subset of X. Proposition 9.4.2 (linear selection) Let A : X ⇒ X be a maximally monotone linear relation. Then the following hold. (i) (∀x ∈ dom A) QA x = P(A0)⊥ (Ax), and QA x ∈ Ax. (ii) QA is monotone and linear. (iii) A = QA + A0. Proof. Let x ∈ dom A = dom QA and let x∗ ∈ Ax. Using Proposition 3.1.3(ii), we see that QA x = PAx x = Px∗ +A0 x = x∗ + PA0 (x − x∗ ) = x∗ + PA0 x − PA0 x∗ = PA0 x + P(A0)⊥ x∗ = P(A0)⊥ x∗ . 218 9.4. When X is a Hilbert space Since x∗ ∈ Ax is arbitrary, we have thus established (i). Now let x and y be in dom A, and let α and β be in R. If α = β = 0, then, by Proposition 3.1.3(i), we have QA (αx + βy) = QA 0 = PA0 0 = 0 = αQA x + βQA y. Now assume that α = 0 or β = 0. By (i) and Proposition 3.1.3(iii), we have QA (αx + βy) = P(A0)⊥ A(αx + βy) = αP(A0)⊥ (Ax) + βP(A0)⊥ (Ay) = αQA x + βQA y. Hence QA is a linear selection of A and (ii) holds. Finally, (iii) follows from Proposition 3.1.3(ii). Example 9.4.3 Let A : X ⇒ X be maximally monotone and skew. Then A = ∂ιdom A + QA is a Borwein-Wiersma decomposition. Proof. By Proposition 9.4.2(ii), QA is a linear selection of A. Now apply Theorem 9.1.12. Example 9.4.4 Let A : X ⇒ X be a maximally monotone linear relation such that dom A is closed. Set B = Pdom A QA Pdom A and f = qB + ιdom A . Then the following hold. (i) B : X → X is continuous, linear, and maximally monotone. (ii) f : X → ]−∞, +∞] is convex, lower semicontinuous, and proper. (iii) A = ∂ιdom A + B. (iv) ∂f + B◦ is a Borwein-Wiersma decomposition of A. 219 9.4. When X is a Hilbert space Proof. (i): By Proposition 9.4.2(ii), QA is monotone and a linear selection of A. Hence, B : X → X is linear; moreover, (∀x ∈ X) x, Bx = x, Pdom A QA Pdom A x = Pdom A x, QA Pdom A x ≥ 0. Altogether, B : X → X is linear and monotone. By Corollary 3.2.3, B is continuous and maximally monotone. (ii): By (i), qB is thus convex and continuous; in turn, f is convex, lower semicontinuous, and proper. (iii): Using Proposition 9.4.2(i) and Proposition 3.2.2(iii), we have (∀x ∈ X) (QA Pdom A )x ∈ (A0)⊥ = dom A = dom A. Hence, (∀x ∈ dom A) Bx = (Pdom A QA Pdom A )x = QA x ∈ Ax. Thus, B + Idom A = QA . In view of Proposition 9.4.2(iii) and Proposition 3.2.2(iii), we now obtain A = B + Idom A + A0 = B + ∂ιdom A . (iv): It follows from (iii) and (9.2) that A = B + ∂ιdom A = ∇qB + ∂ιdom A + B◦ = ∂(qB + ιdom A ) + B◦ = ∂f + B◦ . Proposition 9.4.5 Let A : X ⇒ X be such that dom A is a closed subspace of X. Then A is a maximally monotone linear relation ⇔ A = ∂ιdom A + B, where B : X → X is linear and monotone. Proof. “⇒”: This is clear from Example 9.4.4(i)&(iii). “⇐”: Clearly, A is a linear relation. By Corollary 3.2.3, B is continuous and maximally monotone. Using Rockafellar’s sum theorem [66] or Theorem 5.3.1, we conclude that ∂ιdom A + B is maximally monotone. 220 9.5. Discussion 9.5 Discussion The original papers by Asplund [1] and by Borwein and Wiersma [27] concerned the additive decomposition of a maximally monotone operator whose domain has nonempty interior. In this chapter, we focused on maximally monotone linear relations and we specifically allowed for domains with empty interior. All maximally monotone linear relations on finitedimensional spaces are Borwein-Wiersma decomposable; however, this fails in infinite-dimensional settings. We presented characterizations of BorweinWiersma decomposability of maximally monotone linear relations in general Banach spaces and provided a more explicit decomposition in Hilbert spaces. The characterization of Asplund decomposability and the corresponding construction of an Asplund decomposition remain interesting unresolved topics for future explorations, even for maximally monotone linear operators whose domains are proper dense subspaces of infinite-dimensional Hilbert spaces. 221 Chapter 10 Conclusion Let us conclude by listing our findings of all relevant chapters. Chapter 3: The Brezis-Browder Theorem (see Fact 3.2.13) is a very important characterization of maximal monotonicities of monotone relations. The original proof [30] is based on the application of Zorn’s Lemma by constructing a series of finite-dimensional subspaces, which is complicated. In Theorem 3.2.15, we establish the Brezis-Browder Theorem by considering the fact that a lower semicontinuous, convex and coercive function on a reflexive space has at least one minimizer. In [75], Simons generalized the Brezis-Browder Theorem to SSDB spaces. The Brezis-Browder Theorem and Corollary 3.2.6 are essential tools for the construction of maximally monotone linear subspace extensions of a monotone linear relation. There will be an interesting question for the future work on the BrezisBrowder Theorem in a general Banach space: Let A : X ⇒ X ∗ be a monotone linear relation such that gra A is closed. Assume A∗ |X is monotone. Is A necessarily maximally monotone? In Sections 3.3 and 3.4, some explicit monotone linear relations were constructed in Hilbert spaces, which gave a negative answer to a question 222 Chapter 10. Conclusion raised by Svaiter [80] and which showed that the constraint qualification in the sum problem for maximally monotone operators cannot be weakened (see [63, Example 7.4]). In particular, these two sections will provide concrete examples for the characterization of decomposable monotone linear relations. Chapter 4: A direction for future work in this chapter is to write computer code to find the maximally monotone subspace extension of G, and to generalize the results into a Hilbert space by applying the Brezis-Browder Theorem. Chapter 5: As we can see, Fact 5.1.7 plays an important role in the proof of Theorem 5.2.4 and Theorem 5.3.1. Theorem 5.2.4 presents a powerful sufficient condition for the sum problem. The following question posed by Simons in [72, Problem 41.4] remains open: Let A : X ⇒ X ∗ be maximally monotone of type (FPV), let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C = ∅. Is A + NC necessarily maximally monotone? If the above result holds, by Theorem 5.2.4, we can get the following result: Let A : X ⇒ X ∗ be maximally monotone of type (FPV), and let B : X ⇒ X ∗ be maximally monotone with dom A ∩ int dom B = ∅. Assume that dom A ∩ dom B ⊆ dom B. Then A + B is maximally monotone. Chapter 6: Our first main result (Theorem 6.2.1) in this chapter is obtained by applying Goldstine’s Theorem (see Fact 6.1.2). Simons, Marques Alves and Svaiter’s characterization of type (D) operators and Borwein’s 223 Chapter 10. Conclusion generalization of the Brøndsted-Rockafellar theorem are the main tools for obtaining the other main result (Theorem 6.3.1). Corollary 6.3.3 motivates the following question: Let A : X ⇒ X ∗ be a monotone linear relation with closed graph. Assume that A∗ is monotone. Is A necessarily of type (D)? Chapter 7: It would be interesting to find out whether Theorem 7.3.1 generalizes to the following: Let A : X ⇒ X ∗ be a maximally monotone linear relation, let C be a nonempty closed convex subset of X. Assume that dom A − λC is a closed subspace of X. λ>0 Is it necessarily true that FA+NC = FA 2 FNC ? Chapter 8: The idea of the construction of the operator A in (Theorem 8.2.2) comes from [4, Theorem 5.1] by Bauschke and Borwein. The main tool involved in the main result (Theorem 8.2.2) is Simons and Z˘ alinescu’s version of Attouch-Brezis theorem. Chapter 9: The original papers by Asplund [1] and by Borwein and Wiersma [27] concerned the additive decomposition of a maximally monotone operator whose domain has nonempty interior. In this chapter, we focused on maximally monotone linear relations and we specifically allowed for domains with empty interior. All maximally monotone linear relations on finite-dimensional spaces are Borwein-Wiersma decomposable; however, 224 Chapter 10. 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Yao, “The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone”, to appear Set-Valued and Variational Analysis; http://arxiv.org/abs/1010.4346v1. 237 Bibliography [92] C. Z˘ alinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, 2002. [93] E. Zeidler, Nonlinear Functional Analysis and its Application, Vol II/B Nonlinear Monotone Operators, Springer-Verlag, New York-BerlinHeidelberg, 1990. 238 Appendix A Maple code The following is the Maple code to plot Figure 2.1. > r e s t a r t : Loading S tu d en t:− L i n e a r A l g e b r a with ( p l o t s ) : > f i e l d p l o t ( ( Matrix ( 2 , 2 , { ( 1 , 1) = 0 , ( 1 , 2) = −1, ( 2 , 1) = 1 , ( 2 , 2) = 0 } ) ) . ( Vector ( 2 , { ( 1 ) = x , ( 2 ) = y } ) ) , x = −3 . . 3 , y = −2 . . 2 , th i ck n es s = 2 , colour = blue ) 239 Appendix A. Maple code The following is the Maple code used to verify the calculations for Example 4.5.2 on G2 > r e s t a r t : Loading S tu d en t:− L i n e a r A l g e b r a >A := Matrix ( 3 , 2 , { ( 1 , 1) = −1, ( 1 , 2) = 0 , ( 2 , 1) = 0 , ( 2 , 2) = 0 , ( 3 , 1) = 0 , ( 3 , 2) = −1}); >B := Matrix ( 3 , 2 , { ( 1 , 1) = 1 , ( 1 , 2) = 0 , ( 2 , 1) = 0 , ( 2 , 2) = 1 , ( 3 , 1) = 0 , ( 3 , 2) = 1}) >T:=A. T r an s p os e (B)+B . T r an s p os e (A) >E i g e n v a l u e s (T) >E i g e n v e c t o r s (T) >Idlam :=[[[ −1+ s q r t ( 2 ) , 0 , 0 ] , [ 0 , − 1 − s q r t ( 2 ) , 0 ] , [ 0 , 0 , − 2 ] ] ] >V := Matrix ( 3 , 3 , { ( 1 , 1) = 0 , ( 1 , 2) = 0 , ( 1 , 3) = 1 , ( 2 , 1) = −1/( s q r t (2) − 1) , ( 2 , 2) = −1/(−1− s q r t ( 2 ) ) , ( 2 , 3) = 0 , ( 3 , 1) = 1 , ( 3 , 2) = 1 , ( 3 , 3) = 0}) >N:= Matrix ( 3 , 3 , { ( 1 , 1) = 0 , ( 1 , 2) = −1, ( 1 , 3) = 1 , ( 2 , 1) = 0 , ( 2 , 2) = 2 , ( 2 , 3) = −1, ( 3 , 1) = 0 , ( 3 , 2) = 1 , ( 3 , 3) = 1}) >M:= T r an s p os e (N ) . Idlam .N > e v a l f ( E i g e n v a l u e s (M) ) >Nu llS p ace ( ‘ <| > ‘( T r an s p os e (N ) . T r an s p os e (V ) . A, T r an s p os e (N) . T r an s p os e (V) . B) ) >C := Matrix ( 2 , 2 , { ( 1 , 1) = 1 , ( 1 , 2) = −2∗ s q r t ( 2 ) , ( 2 , 1) = 0 , ( 2 , 2) = 5∗ s q r t ( 2 ) } ) > t i l d e {G 2}:= 1/C 240 Appendix A. Maple code The following is the Maple code used to verify the calculations for Example 4.5.3 on G1 , G2 , E1 and E2 . > r e s t a r t : Loading S tu d en t:− L i n e a r A l g e b r a v o i s e i >A := Matrix ( [ [ 1 , 1 ] , [ 2 , 0 ] , [ 3 , 1 ] ] ) >B := Matrix ( [ [ 1 , 5 ] , [ 1 , 7 ] , [ 0 , 2 ] ] ) >K := Matrix ( [ [ 1 , 1 , 1 , 5 ] , [ 2 , 0 , 1 , 7 ] , [ 3 , 1 , 0 , 2 ] ] ) >Rank (K) >K1:= A. T r an s p os e (B)+B . T r an s p os e (A) >E i g e n v e c t o r s (K1) >Idlam := Matrix ( 3 , 3 , { ( 1 , 1) = 13+ s q r t ( 2 0 1 ) , ( 1 , 2) = 0 , ( 1 , 3) = 0 , ( 2 , 1) = 0 , ( 2 , 2) = −6, ( 2 , 3) = 0 , ( 3 , 1) = 0 , ( 3 , 2) = 0 , ( 3 , 3) = 13− s q r t ( 2 0 1 ) } ) >V:= Matrix ( 3 , 3 , { ( 1 , 1) = 20/(1+ s q r t ( 2 0 1 ) ) , ( 1 , 2) = 0 , ( 1 , 3) = 20/(1 − s q r t ( 2 0 1 ) ) , ( 2 , 1) = 1 , ( 2 , 2) = −1, ( 2 , 3) = 1 , ( 3 , 1) = 1 , ( 3 , 2) = 1 , ( 3 , 3) = 1}) >V g := Matrix ( 2 , 3 , { ( 1 , 1) = 0 , ( 1 , 2) = −1, ( 1 , 3) = 1 , ( 2 , 1) = 20/(1 − s q r t ( 2 0 1 ) ) , ( 2 , 2) = 1 , ( 2 , 3) = 1}) >L := Nu llS p ace ( ‘ <| > ‘( V g . A, V g . B) ) >C0 := Matrix ( 2 , 2 , { ( 1 , 1) = −(−21+ s q r t (201))/( −2+2∗ s q r t ( 2 0 1 ) ) , ( 1 , 2) = −(−107+7∗ s q r t (201))/( −2+2∗ s q r t ( 2 0 1 ) ) , ( 2 , 1) = (−23+3∗ s q r t (201))/( −2+2∗ s q r t ( 2 0 1 ) ) , ( 2 , 2) = (−117+17∗ s q r t (201))/( −2+2∗ s q r t ( 2 0 1 ) ) } ) 241 Appendix A. Maple code > t i l d e {G 1 }:= 1/C0 >N := Matrix ( 3 , 3 , { ( 1 , 1) = 0 , ( 1 , 2) = 0 , ( 1 , 3) = 1/5 , ( 2 , 1) = 0 , ( 2 , 2) = 1 , ( 2 , 3) = 0 , ( 3 , 1) = 0 , ( 3 , 2) = 0 , ( 3 , 3) = 1}) >M := T r an s p os e (N ) . Idlam .N >e v a l f ( E i g e n v a l u e s (M) ) >Nu llS p ace ( ‘ <| > ‘( T r an s p os e (N ) . T r an s p os e (V ) . A, T r an s p os e (N) . T r an s p os e (V) . B) ) >C1 := Matrix ( 2 , 2 , { ( 1 , 1) = −9/20+(1/30)∗ s q r t ( 2 0 1 ) , ( 1 , 2) = −13/4+(1/6)∗ s q r t ( 2 0 1 ) , ( 2 , 1) = 29/20 −(1/30)∗ s q r t ( 2 0 1 ) , ( 2 , 2) = 33/4 −(1/6)∗ s q r t ( 2 0 1 ) } ) > t i l d e {G 2}:= 1/C1 >vec := Vector ( 3 , { ( 1 ) = 0 , ( 2 ) = 0 , ( 3 ) = 0}) >L i n e a r S o l v e ( ‘ <| > ‘(A, B, vec ) , f r e e = t ) 242 Index ε–subdifferential operator, 8, 141 duality mapping, 121 adjoint, 5, 27, 28, 147, 153 Fenchel conjugate, 7 Asplund decomposition, 200, 203 Fitzpatrick function, 6, 101 Attouch & Brezis’ Theorem, 15 Fitzpatrick, Phelps & Veronas’ Theorem, 103 BC–function, 4, 181, 182, 187, 193 Borwein’s Theorem, 141 graph, 5 Borwein-Wiersma decomposition, 199, 211, 212, 214 boundary, 7 Brezis & Browder’s Theorem, 27 identity mapping, 167 indicator function, 7 indicator mapping, 7 inf-convolution, 8, 15 Censor, Iusem & Zenios’ Theorem, 200 interior, 7 inverse operator, 5 closed unit ball, 8 irreducible, 199 constraint qualification, 99 isometric, 182 convex hull, 7 isometry, 182 Crouzeix & Oca˜ na-Anaya’s char- isomorphism into, 182 acterizations, 70 linear relation, 6 distance function, 7 lower semicontinuous hull, 7 domain, 5 243 Index maximally monotone, 6 skew part, 19 maximally skew, 40 subdifferential operator, 8, 100, 102, maximally skew extension, 40 103, 141 monotone, 6 sum operator, 99 monotonically related to, 6 sum problem, 99, 115, 122 norm closure, 7 symmetric, 19 symmetric part, 19 open unit ball, 8 type (D), 140, 146, 153, 184 paramonotone, 200 type (FPV), 6, 103, 115, 133 partial inf-convolution, 156, 159, type Fitzpatrick-Phelps (FP), 140, 182 141, 143, 146, 153 range, 5 type Fitzpatrick-Phelps-Veronas, 6 representative, 140, 142 type negative infimum (NI), 140, right and left shift operator, 39 143, 146, 147, 153 Rockafellar’s Theorems, 99, 100 Voisei’s Theorem, 103 set-valued operator, 5 Volterra integration operator, 43 Simons & Veronas’ Theorem, 103 weak closure, 7 Simons & Z˘ alinescu’s Theorems, 28, 159, 182 Simons’ Theorems, 101–103, 141, 183 Simons, Marques Alves & Svaiter’s Theorem, 142 skew, 19 244
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On Monotone linear relations and the sum problem in Banach spaces Yao, Liangjin 2011
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Title | On Monotone linear relations and the sum problem in Banach spaces |
Creator |
Yao, Liangjin |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | We study monotone operators in general Banach spaces. Properties and characterizations of monotone linear relations are presented. We focus on the "sum problem" which is the most famous open problem in Monotone Operator Theory, and we provide a powerful sufficient condition for the sum problem. We work on classical types of maximally monotone operators and provide affirmative answers to several open problems posed by Phelps and by Simons. Borwein-Wiersma decomposition and Asplund decomposition of maximally monotone operators are also studied. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0072528 |
URI | http://hdl.handle.net/2429/39970 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Irving K. Barber School of Arts and Sciences (Okanagan) Mathematics, Department of (Okanagan) |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-05 |
Campus |
UBCO |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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