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On Monotone linear relations and the sum problem in Banach spaces 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 Created | 2012-01-10 |
Date Issued | 2012-01-10 |
Date | 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. |
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Thesis/Dissertation |
Type |
Text |
Language | Eng |
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Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2012-01-10 |
DOI | 10.14288/1.0072528 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-05 |
Campus |
UBCO |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/39970 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0072528/source |
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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]; Chap- ter 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 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Notation and examples . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 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 rela- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1 Auxiliary results on linear relations . . . . . . . . . . . . . . 59 4.1.1 One linear relation: two equivalent formulations . . . 65 4.2 Explicit maximally monotone extensions of monotone linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Minty parameterizations . . . . . . . . . . . . . . . . . . . . 79 4.4 Maximally monotone extensions with the same domain or the same range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5 The sum problem . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Maximality of the sum of a (FPV) operator and a full domain operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3 Maximality of the sum of a linear relation and a subdifferen- tial operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.4 An example and comments . . . . . . . . . . . . . . . . . . . 135 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6 Classical types of maximally monotone operators . . . . . 139 6.1 Introduction and auxiliary results . . . . . . . . . . . . . . . 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 convo- lution 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 . . . . . . . . . . . . . . . . . . . . . . 178 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . 192 8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9 On Borwein-Wiersma decompositions of monotone linear re- lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 9.1 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.2 Uniqueness results . . . . . . . . . . . . . . . . . . . . . . . . 207 9.3 Characterizations and examples . . . . . . . . . . . . . . . . 214 9.4 When X is a Hilbert space . . . . . . . . . . . . . . . . . . . 218 9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 vi Table of Contents 10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Appendices A Maple code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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+ 1 2A ∗ p. 19 A◦ 12A− 12A∗ p. 19 BX the closed unit ball of X p. 8 D⊥ { z ∈ Z | 〈z, d∗〉 = 0, ∀d∗ ∈ D} p. 5 F ᵀ p. 159 F11F2 p. 181 F12F2 the partial inf-convolution of F1 and F2 p. 156 FA 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) 7→ x p. 8 PY X × Y → Y : (x, y) 7→ y p. 8 QA p. 218 ix List of Symbols S–saturated p. 182 S⊥ { z∗ ∈ Z∗ | 〈z∗, s〉 = 0, ∀s ∈ S} p. 5 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 ΦA p. 143 bdryC the boundary of C p. 7 convC the convex hull of C p. 7 d(·, C) the distance function to a set C p. 7 dimF the dimension of F p. 7 domA the domain of A p. 5 dom f f−1(R) p. 7 `2(N) p. 30 graA the graph of A p. 5 intC the interior of C p. 7 C∗w* the weak∗ closure of C∗ p. 7 C∗∗w* the weak∗ closure of C∗∗ p. 7 C the norm closure of C p. 7 C w the weak closure of C p. 7 f the lower semicontinuous hull of f p. 7 ∂εf the ε–subdifferential operator of f p. 8 ranA the range of A p. 5 Sgn p. 8 x List of Symbols σC the support function of C p. 7 fg the inf-convolution of f and g p. 8 f ⊕ g p. 8 f∗ the Fenchel conjugate of f p. 7 qA 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 influ- enced 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 characteri- zations 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 mono- tone linear relations, which generalize Crouzeix and Anaya’s recent work. The most important open problem in Monotone Operator Theory con- cerns 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 Chap- ter 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 subdiffer- ential operator. In Chapter 6, we study classical types of maximally monotone opera- tors: 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 inf- convolution 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 op- erators: Asplund Decomposition and Borwein-Wiersma Decomposition. In Chapter 9, we show that Borwein-Wiersma decomposability implies Asplund decomposability. We present characterizations of Borwein-Wiersma decom- posability 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 : domA→ X∗ be linear and maximally monotone, let C be a nonempty closed convex subset of X, and suppose that domA ∩ intC 6= ∅. Is A+NC necessarily maximally monotone? (2) Simons posed the following question in [74, Problem 47.6] (See Theo- rem 6.2.1 or [6].): Let A : domA → 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 : domA → 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 Exam- ple 8.3.1(iii)&(v) or [8].): Let F1, F2 : X × X∗ → ]−∞,+∞] be proper lower semicon- tinuous and convex functions. Assume that F1, F2 are BC– functions and that ⋃ λ>0 λ [PX∗ domF1 − PX∗ domF2] is a closed subspace of X∗. Is F11F2 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 graA = {(x, x∗) ∈ X ×X∗ | x∗ ∈ Ax} be the graph of A. The inverse operator A−1 : X∗ ⇒ X is given by graA−1 = { (x∗, x) ∈ X∗ ×X | x∗ ∈ Ax}; the domain of A is domA ={ x ∈ X | Ax 6= ∅}, and its range is ranA = A(X). If Z is a real Banach space with dual Z∗ and a set S ⊆ Z, we define S⊥ by S⊥ = {z∗ ∈ Z∗ | 〈z∗, s〉 = 0, ∀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 graA∗ = { (x∗∗, x∗) ∈ X∗∗ ×X∗ | (x∗,−x∗∗) ∈ (graA)⊥} = { (x∗∗, x∗) ∈ X∗∗ ×X∗ | 〈x∗, a〉 = 〈a∗, x∗∗〉, ∀(a, a∗) ∈ graA}. 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∗ 7→ sup (a,a∗)∈graA (〈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∗) ∈ graA)(∀(y, y∗) ∈ graA) 〈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 graA if 〈x− y, x∗ − y∗〉 ≥ 0, ∀(y, y∗) ∈ graA. Let A : X ⇒ X∗ be maximally monotone. We say A is of type Fitzpatrick- Phelps-Veronas (FPV) if for every open convex set U ⊆ X such that U ∩ domA 6= ∅, the implication x ∈ U and (x, x∗) is monotonically related to graA ∩ (U ×X∗) ⇒ (x, x∗) ∈ graA holds. We say A is a linear relation if graA 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, intC is the interior of C, bdryC is the boundary of C, convC is the convex hull of C, and C and C w are respectively the norm closure of C and weak closure of C. For the set C∗ ⊆ X∗, C∗w* 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; ∞, otherwise. (2.3) The indicator mapping IC : X → X∗ is defined by IC(x) = 0, if x ∈ C; ∅, otherwise. (2.4) The distance function to the set C, written as d(·, C), is defined by x 7→ infc∈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 dimF stand for the dimension of a subspace F of X. Given f : X → ]−∞,+∞], we set dom f = f−1(R) and f∗ : X∗ → [−∞,+∞] : x∗ 7→ 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 6= 7 Chapter 2. Notation and examples ∅, then ∂f : X ⇒ X∗ : x 7→ {x∗ ∈ X∗ | (∀y ∈ X) 〈y − x, x∗〉+ f(x) ≤ f(y)} is the subdifferential operator of f . Note that NC = ∂ιC For ε ≥ 0, the ε–subdifferential of f is defined by ∂εf : X ⇒ X ∗ : x 7→ {x∗ ∈ X∗ | (∀y ∈ X) 〈y − x, x∗〉+ f(x) ≤ f(y) + ε}. We have ∂f = ∂0f . Let g : X → ]−∞,+∞]. The inf-convolution of f and g, fg, is defined by fg : x 7→ inf y∈X [f(y) + g(x− y)] . Let J be the duality map, i.e., the subdifferential of the function 12‖ · ‖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) 7→ x, and PY : X×Y → Y : (x, y) 7→ 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 = { x ∈ X | ‖x‖ < 1}, the closed unit ball in X is denoted by BX = { x ∈ X | ‖x‖ ≤ 1} and N = {1, 2, 3, . . .}. Let Sgn be defined by Sgn: R⇒ R : ξ 7→ 1, if ξ > 0; [−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 Ex- ample 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: A = 0 −1 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 = BTx+ S⊥, if x ∈ V ⊥; ∅, otherwise. 9 2.1. Some examples Figure 2.1: field plot of the linear operator A That is, graA = span{(s1, Bs1), . . . , (sp, Bsp), (0, v1), . . . , (0, vq)} graA∗ = span{(v′1, B>v′1), . . . , (v′p′ , B>vp′), (0, s′1), . . . , (0, s′q′)} where, (s1, . . . , sp), (v1, . . . , vq) are respectively the bases of S and V and (v′1, . . . , v ′ p′), (s ′ 1, . . . , 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∗ = BT = 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 Ax = 4 1 −1 1 2 1 −1 1 2 x+ span e1, if x ∈ span{e2}; ∅, otherwise, where e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1). Then A∗x = 4 1 −1 1 2 1 −1 1 2 x+ span{e1, e3}, if x ∈ span{e2, e3}; ∅, otherwise, and graA = span{(0, e1), (e2, e1 + 2e2 + e3)} graA∗ = 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 NBX is maximally monotone; furthermore, for every x ∈ `1(N), NBX (x) = { 0 } , if ‖x‖ < 1; R+ · ( Sgn(xn) ) 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))∞n=1, x〉 = 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) )∞ n=1 | K ≥ 0}. Now assume K := ‖x∗‖∗ 6= 0. Thus, |x∗n| ≤ K, ∀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 6= 0. We can suppose xn > 0. By (2.6), we have K = x∗nxn + ∑ i 6=n x∗ixi ≤ x∗nxn + ∑ i 6=n sup j∈N |x∗j | · |xi| = x∗nxn +K(1− xn) ≤ Kxn +K(1− xn) = K. Hence x∗nxn + K(1 − xn) = Kxn + K(1 − xn). Thus, x∗n = K. Then x∗ ∈ (K · Sgn(xn))∞n=1. That is, NBX (x) ⊆ {( K · Sgn(xn) )∞ n=1 | K ≥ 0}. Hence NBX (x) = {( K · Sgn(xn) )∞ n=1 | 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 rela- tions, present some sufficient conditions for a linear relation to be mono- tone, and construct some examples of maximally monotone linear rela- tions. Furthermore, we provide a brief proof of the Brezis-Browder The- orem on the characterization of the maximal monotonicity of linear rela- tions. 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 re- lations, 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 domA to X∗ and (abusing notation slightly) also write A : domA → X∗. An analogous comment applies conversely to a linear single-valued operator A 14 3.1. Properties of linear relations with domain domA, 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 Theo- rem 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 func- tions. Assume that ⋃ λ>0 λ [dom f − dom g] is a closed subspace of X. Then (f + g)∗(z∗) = min y∗∈X∗ {f∗(y∗) + g∗(z∗ − y∗)}, ∀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 follow- ing hold. (i) A0 is a linear subspace of X∗. (ii) Ax = x∗ +A0, ∀x∗ ∈ Ax. (iii) (∀(α, β) ∈ R2 r {(0, 0)}) (∀x, y ∈ domA) A(αx + βy) = αAx+ βAy. 15 3.1. Properties of linear relations (iv) (A∗)−1 = (A−1)∗. (v) (∀x ∈ domA∗)(∀y ∈ domA) 〈A∗x, y〉 = 〈x,Ay〉 is a singleton. (vi) If X is reflexive and graA is closed, then A∗∗ = A. (vii) (domA)⊥ = A∗0 and domA = (A∗0)⊥. (viii) If graA is closed, then (domA∗)⊥ = A0 and domA∗ w* = (A0)⊥. (ix) If domA is closed, then domA∗ = (Ā0)⊥ and thus domA∗ is (weak∗) closed, where Ā is the linear relation whose graph is the closure of the graph of A. (x) If k ∈ Rr {0}, then (kA)∗ = kA∗. Proof. (i): Since graA is a linear subspace, {0} ×A0 = graA ∩ {0} ×X∗ is a linear subspace and hence A0 is a linear subspace. (ii): Let x ∈ domA and x∗ ∈ Ax. Then (x, x∗+A0) = (x, x∗)+(0, A0) ⊆ graA and hence x∗ + A0 ⊆ Ax. On the other hand, let y∗ ∈ Ax. We have (0, y∗ − x∗) = (x, y∗) − (x, x∗) ∈ graA. Then y∗ − x∗ ∈ A0 and thus y∗ ∈ x∗ +A0. Hence Ax ⊆ x∗ +A0 and thus Ax = x∗ +A0. (iii): Let (α, β) ∈ R2 r {(0, 0)} and {x, y} ⊆ domA. We can suppose α 6= 0 and β 6= 0. Take x∗ ∈ Ax and y∗ ∈ Ay. Since graA is a linear subspace, αx∗+βy∗ ∈ A(αx+βy). By (ii), A(αx+βy) = αx∗+βy∗+A0 = αx∗ +A0 + βy∗ +A0 = α(x∗ + 1 α A0) + β(y∗ + 1 β A0) = αAx+ βAy. (iv): We have (x∗, x∗∗) ∈ gra(A∗)−1 ⇔ (x∗∗, x∗) ∈ graA∗ ⇔ (x∗,−x∗∗) ∈ (graA)⊥ ⇔ (x∗∗,−x∗) ∈ (graA−1)⊥ ⇔ (x∗, x∗∗) ∈ gra(A−1)∗. 16 3.1. Properties of linear relations (v): Let x ∈ domA∗ and y ∈ domA. 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∗) ∈ graA∗∗ ⇔ (x∗,−x) ∈ (graA∗)⊥ = ((gra−A−1)⊥)⊥ = gra−A−1 ⇔ (x, x∗) ∈ graA. (vii): Clearly, (domA)⊥ ⊆ A∗0. Let x∗ ∈ A∗0. We have 〈x∗, y〉 + 〈0, Ay〉 = 0, ∀y ∈ domA. Then we have x∗ ∈ (domA)⊥ and thus A∗0 ⊆ (domA)⊥. Hence (domA)⊥ = A∗0. By Fact 3.1.1, domA = (A∗0)⊥. (viii): By Fact 3.1.1, x∗ ∈ A0⇔ (0, x∗) ∈ graA = [ (graA)⊥ ] ⊥ = [ gra−(A∗)−1]⊥ ⇔ 〈x∗, y∗∗〉 = 0, ∀y∗∗ ∈ domA∗ ⇔ x∗ ∈ (domA∗)⊥. Hence (domA∗)⊥ = A0. Take Y = X∗, by Fact 3.1.1 again, domA∗ w* = (A0)⊥. (ix): Let Ā be the linear relation whose graph is the closure of the graph of A. Then domA = dom Ā and A∗ = Ā∗. Then by Fact 3.1.2, ιX∗×(Ā0)⊥ = ι ∗ {0}×Ā0 = ( ιgra Ā + ι{0}×X∗ )∗ = ιgra(−Ā∗)−1 ιX∗×{0} = ιX∗×dom Ā∗ . It is clear that domA∗ = dom Ā∗ = (Ā0)⊥ is closed. (x): Let k ∈ Rr {0}. Then (x∗∗, x∗) ∈ gra(kA)∗ ⇔ (x∗,−x∗∗) ∈ (gra kA)⊥ ⇔ (x∗,−kx∗∗) ∈ (graA)⊥ ⇔ ( 1 k x∗,−x∗∗) ∈ (graA)⊥ ⇔ (x∗∗, 1 k x∗) ∈ graA∗. 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 domA ⊆ (A0)⊥ and A0 ⊆ (domA)⊥; consequently, if graA is closed, then domA ⊆ domA∗w* ∩ X and A0 ⊆ A∗0. (ii) (∀x ∈ domA)(∀z ∈ (A0)⊥) 〈z,Ax〉 is single-valued. (iii) (∀z ∈ (A0)⊥) domA→ R : y 7→ 〈z,Ay〉 is linear. (iv) A is monotone ⇔ (∀x ∈ domA) 〈x,Ax〉 is single-valued and 〈x,Ax〉 ≥ 0. (v) If (x, x∗) ∈ (domA) × X∗ is monotonically related to graA and x∗0 ∈ Ax, then x∗ − x∗0 ∈ (domA)⊥. Proof. (i): Pick x ∈ domA. Then there exists x∗ ∈ X∗ such that (x, x∗) ∈ graA. By the monotonicity of A and since (0, A0) ⊆ graA, we have 〈x, x∗〉 ≥ sup〈x,A0〉. Since A0 is a linear subspace (Proposition 3.1.3(i)), we obtain x⊥A0. This implies domA ⊆ (A0)⊥ and A0 ⊆ (domA)⊥. If graA is closed, then Proposition 3.1.3(viii)&(vii) yield domA ⊆ (A0)⊥ ⊆ (A0)⊥ = domA∗w* and A0 ⊆ A∗0. 18 3.2. Properties of monotone linear relations (ii): Take x ∈ domA, 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 ∈ domA) 〈z,Ay〉 is single-valued. Now let x, y be in domA, 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) 6= (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) ∈ graA. “⇐”: If x and y belong to domA, then Proposition 3.1.3(iii) yields 〈x − y,Ax−Ay〉 = 〈x− y,A(x− y)〉 ≥ 0. (v): Let (x, x∗) ∈ (domA)×X∗ be monotonically related to graA, and take x∗0 ∈ Ax. For every (v, v∗) ∈ graA, 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〈domA, x∗ − x∗0〉. Since domA is linear, it follows that x∗ − x∗0 ∈ (domA)⊥. We say that a linear relation A : X ⇒ X∗ is skew if graA ⊆ gra(−A∗), equivalently, if 〈x, x∗〉 = 0, ∀(x, x∗) ∈ graA; furthermore, A is symmetric if graA ⊆ graA∗; equivalently, if 〈x, y∗〉 = 〈y, x∗〉, ∀(x, x∗), (y, y∗) ∈ graA. We define the symmetric part and the skew part of A via A+ = 1 2A+ 1 2A ∗ and A◦ = 12A− 12A∗, (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 (domA)⊥ = A0 and hence domA = (A0)⊥. (ii) If domA is closed, then: A is maximally monotone⇔ (domA)⊥ = A0. (iii) If A is maximally monotone, then domA∗w* ∩ X = domA = (A0)⊥ and A0 = A∗0 = A+0 = A◦0 = (domA)⊥. (iv) If A is maximally monotone and domA is closed, then domA∗ ∩X = domA. (v) If A is maximally monotone and domA ⊆ domA∗, then A = A++A◦, A+ = A−A◦ , and A◦ = A−A+. Proof. (i): Since A + NdomA = A + (domA) ⊥ is a monotone extension of A and A is maximally monotone, we must have A + (domA)⊥ = A. Then A0 + (domA)⊥ = A0. As 0 ∈ A0, (domA)⊥ ⊆ A0. The reverse inclusion follows from Proposition 3.2.1(i). Then we have (domA)⊥ = A0. By Fact 3.1.1, domA = (A0)⊥. (ii): “⇒”: This follows directly from (i). “⇐”: By our assumptions and Fact 3.1.1, domA = (A0)⊥. Let (x, x∗) be monotonically related to graA. We have inf [〈x− 0, x∗ −A0〉] ≥ 0. Then we have x ∈ (A0)⊥ and hence x ∈ domA. 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 = (domA)⊥ = A∗0 and thus A+0 = A◦0 = A0 = (domA)⊥. Then by Proposition 3.1.3(viii) and (i), 20 3.2. Properties of monotone linear relations domA∗w* ∩X = (A0)⊥ = domA. (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, domA+ = domA◦ = domA ∩ domA∗ = domA. Let x ∈ domA, 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 Proposi- tion 3.2(h)] by Phelps and Simons. Voisei and Zălinescu 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 graA is closed. By the Closed Graph Theorem, A is continuous. Proposition 3.2.2(ii) provides a characterization of maximal monotonic- ity for certain monotone linear relations. More can be said in finite- dimensional spaces. We require the following lemma, where dimF 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(graA) = dim(domA) + dimA0. Proof. We shall construct a basis of graA. 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 domA. 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 graA, where x∗i ∈ Axi, i ∈ {k + 1, . . . , l}. Thus dim(graA) = l = dim(domA) + dimA0. 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 = dimX, and let A : X ⇒ X∗ be a monotone linear relation. Then A is maximally monotone if and only if dimgraA = n. Proof. Since linear subspaces ofX are closed, we see from Proposition 3.2.2(ii) that A is maximally monotone⇔ domA = (A0)⊥. (3.3) Assume first that A is maximally monotone. Then domA = (A0)⊥. By Lemma 3.2.4, dim(graA) = dim(domA)+dim(A0) = dim((A0)⊥)+dim(A0) = n. Conversely, let dim(graA) = n. By Lemma 3.2.4, we have that dim(domA) = n − dim(A0). As dim((A0)⊥) = n − dim(A0) and domA ⊆ (A0)⊥ by Proposition 3.2.1(i), we have that domA = (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 fol- lowing are equivalent: (i) A is maximally monotone. 22 3.2. Properties of monotone linear relations (ii) dimgraA = n. (iii) domA = (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 2〈x,Ax〉, if x ∈ domA; ∞, otherwise. Proposition 3.2.7 Let A : X ⇒ X∗ be a monotone linear relation, let x and y be in domA, 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) = 12λ(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 domA 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 graA 6= ∅. Then the function g : X ×X∗ → ]−∞,+∞] : (x, x∗) 7→ 〈x, x∗〉+ ιgraA(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 graA 6= ∅. To see that g is convex, let (a, a∗) and (b, b∗) be in graA, and let λ ∈ ]0, 1[. Set µ = 1−λ ∈ ]0, 1[ and observe that λ(a, a∗) + µ(b, b∗) = (λa+ µb, λa∗ + µb∗) ∈ graA by convexity of graA. 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 rela- tion. Then (x, x∗) ∈ X ×X∗ is monotonically related to graA if and only if 〈x, x∗〉 ≥ 0 and [〈y∗, x〉+ 〈x∗, y〉]2 ≤ 4〈x∗, x〉〈y∗, y〉, ∀(y, y∗) ∈ graA. Proof. We have (x, x∗) ∈ X ×X∗ is monotonically related to graA 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∗) ∈ graA ⇔ 〈x, x∗〉 ≥ 0 and [〈y∗, x〉+ 〈x∗, y〉]2 ≤ 4〈x∗, x〉〈y∗, y〉, ∀(y, y∗) ∈ graA (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 + ιdomA+ = qA+ and thus qA+ is convex. (ii) graA+ ⊆ 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 domA is closed, then A∗|X is maxi- mally monotone. Proof. Let x ∈ domA+. (i): Since A is monotone, by Proposition 3.1.3(v) and Proposition 3.2.1(iv), qA+ = qA|domA+ and A+ is monotone. Then by Propo- sition 3.2.7, qA+ is convex. Let y ∈ domA. Then by Proposition 3.1.3(v) again, 0 ≤ 12〈Ax−Ay, x− y〉 = 12 〈Ay, y〉+ 12〈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 ∈ domA. 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 = domA, by (3.7), qA(z) ≥ qA(x) + 〈A+x, z − x〉, ∀z ∈ dom qA. Hence A+x ⊆ ∂qA(x). If A+ is maximally monotone, then A+ = ∂qA. Thus (ii) holds. (iii): Suppose to the contrary that A∗|X is not monotone. By Proposi- tion 3.2.1(iv), there exists (x0, x ∗ 0) ∈ graA∗ 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∗) ∈ graA. (3.8) Thus, (−x0, x∗0) is monotonically related to graA. By the maximal mono- tonicity of A, (−x0, x∗0) ∈ graA. Then 〈−x0 − (−x0), x∗0 − x∗0〉 = 0, which contradicts (3.8). Hence A∗|X is monotone. (iv): By Proposition 3.1.3(ix), domA∗|X = (A0)⊥ and thus domA∗|X is closed. By Fact 3.1.1 and Proposition 3.2.2(i), (domA∗|X)⊥ = ((A0)⊥)⊥ = A0 w ∗ = A0. Then by Proposition 3.2.2(iii), (domA∗|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 re- lation. Then A is symmetric ⇔ A = A∗|X . Proof. “⇒”: Assume that A is symmetric, i.e., graA ⊆ graA∗. 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 : domA→ X∗ be monotone and linear. The following hold. (i) If A is maximally monotone, then domA is dense (and hence A∗ is at most single-valued). (ii) Assume that A is skew such that domA is dense. Then domA ⊆ domA∗ and A∗|domA = −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 graA 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-Zălinescu) (See [77, Theorem 1.2] or [72, Theo- rem 10.6].) Assume X is reflexive. Let A : X ⇒ X∗ be monotone. Then A is maxi- mally monotone if and only if graA+ 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∗ ⊆ graA + gra(−J). For this, let (x, x∗) ∈ X ×X∗ and we define g : X ×X∗ → ]−∞,+∞] by (y, y∗) 7→ 12‖y∗‖2 + 12‖y‖2 + 〈y∗, y〉+ ιgraA(y − x, y∗ − x∗). We have f : (y, y∗) 7→ 〈y∗, y〉+ιgraA(y−x, y∗−x∗) = 〈y∗, y〉+ιgraA+(x,x∗)(y, y∗). By Fact 3.2.8 and the assumption that graA 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∗) ∈ graA, hence, (x, x∗) ∈ graA+ (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 (z∗0 , z0) ∈ ∂(ιgraA(· − x, · − x∗))(z, z∗) = ∂ιgraA(z − x, z∗ − x∗) = (graA)⊥, and (v, v∗) ∈ X ×X∗ with v∗ ∈ Jz, z∗ ∈ Jv such that (0, 0) = (z∗, z) + (v∗, v) + (z∗0 , z0). Then (− (z + v), z∗ + v∗) ∈ graA∗. Since A∗ is monotone, 〈z∗ + v∗, z + v〉 = 〈z∗, z〉+ 〈z∗, v〉+ 〈v∗, z〉+ 〈v∗, v〉 ≤ 0. (3.10) Note that since 〈z∗, v〉 = ‖z∗‖2 = ‖v‖2, 〈v∗, z〉 = ‖v∗‖2 = ‖z‖2, by (3.10), we have 1 2‖z‖2 + 12‖z∗‖2 + 〈z∗, z〉+ 12‖v∗‖2 + 12‖v‖2 + 〈v, v∗〉 ≤ 0. Hence z∗ ∈ −Jz. By (3.9), (x, x∗) ∈ graA+ 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 `2(N) 3.3 An unbounded skew operator on `2(N) In this section, we construct a maximally monotone and skew operator S on `2(N) 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., domS $ domS∗. 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 (xn)n∈N = (x1, x2, x3, . . .) with ∑ i≥1 x 2 i < +∞. Example 3.3.1 Let H = `2(N), and S : domS → `2(N) be given by Sy = (∑ i<n yi − ∑ i>n yi ) n∈N 2 = (∑ i<n yi + 1 2yn ) n∈N , ∀y = (yn)n∈N ∈ domS, (3.11) where domS = { y = (yn) ∈ `2(N) | ∑ i≥1 yi = 0, (∑ i≤n yi ) n∈N ∈ `2(N)} and ∑ i<1 yi is understood to mean 0. In matrix form, S = 12 0 −1 −1 −1 −1 · · · −1 −1 · · · 1 0 −1 −1 −1 · · · −1 −1 · · · 1 1 0 −1 −1 · · · −1 −1 · · · 1 1 1 0 −1 · · · −1 −1 · · · 1 1 1 1 0 · · · −1 −1 · · · ... . . . . . . . . . . . . , 30 3.3. An unbounded skew operator on `2(N) or S = 1 2 0 0 0 0 · · · 0 0 · · · 1 12 0 0 0 · · · 0 0 · · · 1 1 12 0 0 · · · 0 0 · · · 1 1 1 12 0 · · · 0 0 · · · 1 1 1 1 12 · · · 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 ∈ domS. Then (∑ i≤n xi ) n∈N ∈ `2(N). Thus, `2(N) 3 (∑ i≤n xi ) n∈N − 12x = (∑ i≤n xi ) n∈N − 12(xn)n∈N = (∑ i<n xi + 1 2xn ) n∈N = Sx. Hence S is well defined. Clearly, S is linear on domS. Now we show S is skew. Let y = (yn)n∈N ∈ domS, and s = ∑ i≥1 yi. Then (∑ i≤n yi ) n∈N ∈ `2(N). Hence (∑ i<n yi ) n∈N = (∑ i≤n yi ) n∈N − (yn)n∈N ∈ `2(N). Since s = 0, `2(N) 3 − (∑ i<n yi ) n∈N = 0− (∑ i<n yi ) n∈N = (∑ i≥1 yi − ∑ i<n yi ) n∈N = (∑ i≥n yi ) n∈N , 31 3.3. An unbounded skew operator on `2(N) ( ∑ i≥n+1 yi ) n∈N = 0− (∑ i≤n yi ) n∈N ∈ `2(N). (3.12) Thus, by (3.12), − 2〈Sy, y〉 = 〈(∑ i>n yi − ∑ i<n yi ) n∈N , y 〉 = 〈( ∑ i≥n+1 yi + ∑ i≥n yi ) n∈N , y 〉 (3.13) = 〈(∑ i≥1 yi, ∑ i≥2 yi, . . . ) + (∑ i≥2 yi, ∑ i≥3 yi, . . . ) , y 〉 = 〈(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 n [sy1 + (s− y1)y2 + · · ·+ (s− (y1 + · · · + yn−1))yn]+ lim n [(s− y1)y1 + (s− (y1 + y2))y2 + · · ·+ (s− (y1 + · · ·+ yn))yn] = lim n [s(y1 + · · ·+ yn)− y1y2 − (y1 + y2)y3 − · · · − (y1 + · · · + yn−1)yn]+ [s(y1 + · · ·+ yn)− (y21 + · · ·+ y2n)− y1y2 − · · · − (y1 + · · ·+ yn−1)yn] = lim n [2s(y1 + · · ·+ yn)− (y1 + · · ·+ yn)2] = 2s2 − s2 = s2 = 0. 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, graS is closed. Proof. By Proposition 3.3.2, S is skew. Let (x, x∗) ∈ `2(N) × `2(N) be monotonically related to graS. Write x = (xn)n∈N and x∗ = (x∗n)n∈N. By 32 3.3. An unbounded skew operator on `2(N) Fact 3.2.9, we have 〈Sy, x〉+ 〈x∗, y〉 = 0, ∀y ∈ domS. (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 ∈ domS and Sy = (−12 ,−1, . . . ,−1,−12 , 0, . . .). Then by (3.14), − x∗1 + x∗n − 12x1 − 12xn − n−1∑ i=2 xi = 0⇒ x∗n = x∗1 − 12x1 + n−1∑ i=1 xi + 1 2xn. (3.15) Since x∗ ∈ `2(N) and x ∈ `2(N), we have x∗n → 0, xn → 0. Thus by (3.15), − ∑ i≥1 xi = x ∗ 1 − 12x1. (3.16) Next we show −∑i≥1 xi = x∗1 − 12x1 = 0. Let s =∑i≥1 xi. Then by (3.15) and (3.16), 2x∗ = 2(x∗n)n∈N = 2 ( − ∑ i≥1 xi + ∑ i<n xi + 1 2xn ) n∈N = ( − 2 ∑ i≥1 xi + 2 ∑ i<n xi + xn ) n∈N = ( − 2 ∑ i≥n xi + xn ) n∈N = ( − ∑ i≥n xi − ∑ i≥n xi + xn ) n∈N = ( − ∑ i≥n xi − ∑ i≥n+1 xi ) n∈N . (3.17) 33 3.3. An unbounded skew operator on `2(N) On the other hand, by (3.15) and (3.16), `2(N) 3 x∗ − 12x = ( − ∑ i≥1 xi + ∑ i<n xi + 1 2xn ) n∈N − (12xn)n∈N = ( − ∑ i≥n xi ) n∈N . Then by (3.17), 2x∗ = ( − ∑ i≥n xi ) n∈N + ( − ∑ i≥n+1 xi ) n∈N . Then by Fact 3.2.9, similar to the proof in (3.13) in Proposition 3.3.2, we have 0 ≥ −2〈x∗, x〉 = 〈 (∑ i≥n xi ) n∈N + ( ∑ i≥n+1 xi ) n∈N , x〉 = 〈 (∑ i≥1 xi, ∑ i≥2 xi, . . . ) + (∑ i≥2 xi, ∑ i≥3 xi, . . . ) , x〉 = 2s2 − s2 = s2. Hence s = 0, i.e., x∗1 = 1 2x1 by (3.16). By (3.15), x ∗ = (∑ i<n xi+ 1 2xn ) n∈N . Thus `2(N) 3 x∗ + 12x = (∑ i<n xi + 1 2xn ) n∈N + ( 1 2xn ) n∈N = (∑ i≤n xi ) n∈N . Hence x ∈ domS and x∗ = Sx. Thus, S is maximally monotone. Hence graS is closed. Remark 3.3.4 Let S be as in Example 3.3.1. Since e1 = (1, 0, 0, . . . , 0, . . .) /∈ domS, the operator S is unbounded. 34 3.3. An unbounded skew operator on `2(N) Proposition 3.3.5 Let S be defined as in Example 3.3.1. Then S∗y = (∑ i>n yi + 1 2yn ) n∈N , ∀y = (yn)n∈N ∈ domS∗, (3.18) where domS∗ = { y = (yn)n∈N ∈ `2(N) | ∑ i≥1 yi ∈ R, (∑ i>n yi ) n∈N ∈ `2(N) } . In matrix form, S∗ = 1 2 1 1 1 1 · · · 1 1 · · · 0 12 1 1 1 · · · 1 1 · · · 0 0 12 1 1 · · · 1 1 · · · 0 0 0 12 1 · · · 1 1 · · · 0 0 0 0 12 · · · 1 1 · · · ... . . . . . . . . . . . . . . . · · · · · · . Moreover, domS $ domS∗, S∗ = −S on domS, and S∗ is not skew. Proof. Let y = (yn)n∈N ∈ `2(N) with (∑ i>n yi ) n∈N ∈ `2(N), and y∗ =(∑ i>n yi + 1 2yn ) n∈N . Now we show (y, y∗) ∈ graS∗. Let s =∑i≥1 yi and x ∈ domS. Then we have 〈y, Sx〉 + 〈y∗,−x〉 = 〈 y, 12x+ (∑ i<n xi ) n∈N 〉 + 〈 1 2y + (∑ i>n yi ) n∈N ,−x 〉 = 〈 y, (∑ i<n xi ) n∈N 〉 + 〈(∑ i>n yi ) n∈N ,−x 〉 = lim n [y2x1 + y3(x1 + x2) + · · ·+ yn(x1 + · · ·+ xn−1)] − lim n [x1(s− y1) + x2(s − y1 − y2) + · · ·+ xn(s− y1 − · · · − yn)] 35 3.3. An unbounded skew operator on `2(N) = lim n [x1(y2 + · · ·+ yn) + x2(y3 + · · ·+ yn) + · · · + xn−1yn] − lim n [x1(s− y1) + x2(s − y1 − y2) + · · ·+ xn(s− y1 − · · · − yn)] = lim n [x1(y1 + y2 + · · · + yn − s) + x2(y1 + y2 + · · ·+ yn − s) + · · · + xn(y1 + y2 + · · ·+ yn − s)] = lim n [(x1 + · · ·+ xn)(y1 + y2 + · · ·+ yn − s)] = 0. Hence (y, y∗) ∈ graS∗. On the other hand, let (a, a∗) ∈ graS∗ with a = (an)n∈N and a∗ = (a∗n)n∈N. Now we show (∑ i>n ai ) n∈N ∈ `2(N) and a∗ = (∑ i>n ai + 1 2an ) n∈N . (3.19) Let en = (0, . . . , 0, 1, 0, . . .) : the nth entry is 1 and the others are 0. Then let y = −e1 + en. Thus y ∈ domS and Sy = (−12 ,−1, . . . ,−1,−12 , 0, . . .). Then, 0 = 〈a∗, y〉+ 〈−Sy, a〉 = −a∗1 + a∗n + 12a1 + 12an + n−1∑ i=2 ai ⇒ a∗n = a∗1 − 12a1 − n−1∑ i=2 ai − 12an. (3.20) Since a∗ ∈ `2(N) and a ∈ `2(N), a∗n → 0, an → 0. Thus by (3.20), a∗1 = 1 2a1 + ∑ i>1 ai, (3.21) 36 3.3. An unbounded skew operator on `2(N) from which we see that ∑ i≥1 ai ∈ R. Combining (3.20) and (3.21), we have a∗n = ∑ i>n ai + 1 2an Thus, (3.19) holds. Hence (3.18) holds. Now for x ∈ domS, since ∑i≥1 xi = 0, we have S∗x = ( 1 2xn + ∑ i>n xi ) n∈N = ( − 12xn + ∑ i≥n xi ) n∈N = ( − 12xn − ∑ i<n xi ) n∈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, . . .) ∈ domS∗ but e1 6∈ domS, we have domS $ domS∗. Proposition 3.3.6 Let S be defined as in Example 3.3.1. Then 〈S∗y, y〉 = 12s2, ∀y ∈ domS∗ with s = ∑ i≥1 yi. (3.22) Proof. Let y = (yn)n∈N ∈ domS∗, and s = ∑ i≥1 yi. By Proposition 3.3.5, we have s ∈ R and 〈S∗y, y〉 = 〈 (∑ i>n yi + 1 2yn ) n∈N , y〉 = 〈 (∑ i≥n yi − 12yn ) n∈N , y〉 = lim n [sy1 + (s − y1)y2 + · · ·+ (s− y1 − y2 − · · · − yn−1)yn − 12(y21 + y22 + · · · + y2n)] = lim n [s(y1 + · · ·+ yn)− y1y2 − (y1 + y2)y3 − · · · 37 3.3. An unbounded skew operator on `2(N) − (y1 + y2 + · · ·+ yn−1)yn]− 12 [ y21 + y 2 2 + · · ·+ y2n ] = lim n [s(y1 + · · ·+ yn)] − lim n [y1y2 + (y1 + y2)y3 + · · ·+ (y1 + y2 + · · · + yn−1)yn + 12(y 2 1 + y 2 2 + · · · + y2n)] = s2 − lim n 1 2 [y1 + y2 + · · ·+ yn]2 = s2 − 12s2 = 12s 2. 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 /∈ domS = dom(−S). Thus, (e1, 12e1) /∈ gra(−S). We have for every y ∈ domS, 〈e1, 12e1〉 ≥ 0 and 〈e1,−Sy〉+ 〈y, 12e1〉 = −12y1 + 12y1 = 0. By Fact 3.2.9, (e1, 1 2e1) 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 : domS → X∗ be a maximally monotone skew oper- ator. Then −S has a unique maximally monotone extension: S∗|X . Proof. By Fact 3.2.12, gra(−S) ⊆ graS∗|X . Assume T is a maximally mono- tone extension of −S. Let (x, x∗) ∈ graT . Then (x, x∗) is monotonically related to gra(−S). By Fact 3.2.9, 〈x∗, y〉+ 〈−x, Sy〉 = 〈x∗, y〉+ 〈x,−Sy〉 = 0, ∀y ∈ domS. Thus (x, x∗) ∈ graS∗|X . Since (x, x∗) ∈ graT is arbitrary, we have graT ⊆ graS∗|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 uti- lized 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 de- fined as in Example 3.3.1. Then neither S nor S∗ has full domain. By Fact 3.2.12, ∀x ∈ dom(S + S∗) = domS, 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 graS` = { (x, x∗) ∈ X ×X∗ | (x∗, x) ∈ (graS)⊥}. 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 graT ⊇ graS. 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) ⊆ graT . Thus, graS ⊆ 40 3.3. An unbounded skew operator on `2(N) gra(−T ). Since −T is monotone and S is maximally monotone, graS = 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 mono- tone 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 maxi- mally skew, but −S∗ is not monotone, so not maximally monotone. Proof. Let e1 = (1, 0, 0, . . . , 0, . . .). By Proposition 3.3.5, (e1,−12e1) ∈ gra(−S∗), but 〈e1,−12e1〉 = −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 opti- mal. 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 systemati- cally study T = V −1 and its skew part S = 12(T − T ∗). It turns out that T is neither skew nor symmetric and that its skew part S admits two max- imally monotone and skew extensions T1, T2 (in fact, anti-self-adjoint) even though domS 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 ab- solutely continuous functions on the interval [a, b]. Then ∫ b a xy′ + ∫ b 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 domA = domA∗ and that A has closed graph. Now ∀x ∈ domA, 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 V : H → H : x 7→ V x, where V x : [0, 1]→ R : t 7→ ∫ t 0 x, (3.23) and its adjoint is given by t 7→ (V ∗x)(t) = ∫ 1 t x, ∀x ∈ X. Then (i) Both V and V ∗ are maximally monotone since they are monotone, continuous and linear. (ii) Both ranges ranV = {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 ranV ∗ = {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) ranV ∩ ranV ∗ = {V x | x ∈ e⊥}, where e ≡ 1 ∈ L2[0, 1]. (iv) Define V+x = 1 2(V + V ∗)(x) = 12〈e, x〉e. Then V+ is self-adjoint and ranV+ = span{e}. (v) Define V◦x = 12(V − V ∗)(x) : t 7→ 12 [ ∫ t 0 x − ∫ 1 t x] ∀x ∈ L2[0, 1], t ∈ [0, 1]. Then V◦ is anti-self-adjoint and ranV◦ = {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, 〈x, V x〉 = ∫ 1 0 x(t) ∫ t 0 x(s)dsdt = 1 2 (∫ 1 0 x(s)ds )2 ≥ 0, so V is monotone. As domV = L2[0, 1] and V is continuous, domV ∗ = L2[0, 1]. Let x, y ∈ 44 3.4. The inverse Volterra operator on L2[0, 1] L2[0, 1]. We have 〈V x, y〉 = ∫ 1 0 ∫ t 0 x(s)dsy(t)dt = ∫ 1 0 x(t)dt ∫ 1 0 y(s)ds− ∫ 1 0 ∫ t 0 y(s)dsx(t)dt = ∫ 1 0 (∫ 1 0 y(s)ds − ∫ t 0 y(s)ds ) x(t)dt = ∫ 1 0 ∫ 1 t y(s)dsx(t)dt = 〈V ∗y, x〉, thus (V ∗y)(t) = ∫ 1 t y(s)ds ∀t ∈ [0, 1]. (ii) To show (3.24), if z ∈ ranV , then z(t) = ∫ t 0 x 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 = V z′. To show (3.25), if z ∈ ranV ∗, then z(t) = ∫ 1 t x 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) = 12 (∫ t 0 x′ − ∫ 1 t x′ ) = 12 ( x(t)− x(0)− x(1) + x(t) ) = x(t). This shows that x ∈ ranV◦. Conversely, if x ∈ ranV◦, i.e., x(t) = 1 2 ∫ t 0 y − 1 2 ∫ 1 t y for some y ∈ L2[0, 1], then x is absolutely continuous, x′ = y ∈ L2[0, 1] and x(0) = −x(1) = −12 ∫ 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 = domT ∩ domT ∗. Then the following hold. (i) T : domT → X is given by Tx = x′ with domT = {x ∈ L2[0, 1] : x is absolutely continuous, x(0) = 0, x′ ∈ L2[0, 1]}, and T ∗ : domT ∗ → L2[0, 1] is given by T ∗x = −x′ with domT ∗ = {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 domT = {x ∈ L2[0, 1] : x is absolutely continuous, x(0) = 0, x′ ∈ L2[0, 1]} domT ∗ = {x ∈ L2[0, 1] : x is absolutely continuous, x(1) = 0, x′ ∈ L2[0, 1]} Tx = x′, ∀x ∈ domT, T ∗y = −y′ and ∀y ∈ domT ∗. Note that by Fact 3.4.2, 〈Tx, x〉 = ∫ 1 0 x′x = 1 2 x2(1)− 1 2 x2(0) = 1 2 x(1)2 ∀x ∈ domT, (3.26) 〈T ∗x, x〉 = ∫ 1 0 −x′x = −(1 2 x(1)2 − 1 2 x(0)2) = 1 2 x(0)2 ∀x ∈ domT ∗. (3.27) 47 3.4. The inverse Volterra operator on L2[0, 1] (ii): Letting x(t) = t, y(t) = t2 we have 〈Tx, x〉 = ∫ 1 0 t = 12 , 〈x, Ty〉 = ∫ 1 0 2t2 = 23 6= 13 = ∫ 1 0 t2 = 〈Tx, y〉 ⇒ 〈Tx, x〉 6= 0, 〈Tx, y〉 6= 〈x, Ty〉. (iii): By (i), D = domT∩domT ∗ is clearly a linear subspace. For x ∈ D, x(0) = x(1) = 0, from (3.26) and (3.27), 〈Tx, x〉 = 12x(1)2 = 0, 〈T ∗x, x〉 = 12x(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 = T−T ∗ 2 . (i) Sx = x′ (∀x ∈ domS) and graS = {(V x, x) | x ∈ e⊥}, where e ≡ 1 ∈ L2[0, 1]. In particular, domS = {x ∈ L2[0, 1] : x is absolutely continuous, x(0) = x(1) = 0, x′ ∈ L2[0, 1]}, ranS = {y ∈ L2[0, 1] : 〈e, y〉 = 0} = e⊥. 48 3.4. The inverse Volterra operator on L2[0, 1] Moreover, domS 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 graS∗ = {(V ∗x∗ + le, x∗) | x∗ ∈ L2[0, 1], l ∈ R}. More precisely, S∗x = −x′ ∀x ∈ domS∗, with domS∗ = {x ∈ L2[0, 1] : x is absolutely continuous on [0, 1], x′ ∈ L2[0, 1]}, ranS∗ = L2[0, 1]. Neither S∗ nor −S∗ is monotone. Moreover, S∗∗ = S. (iii) Let T1 : domT1 → L2[0, 1] be defined by T1x = x ′, ∀x ∈ domT1, with domT1 = {x ∈ L2[0, 1] : x is absolutely continuous, x(0) = x(1), x′ ∈ L2[0, 1]}. Then T ∗1 = −T1, ranT1 = 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 domS directly. Now (∀x ∈ domS = domT ∩ domT ∗) Tx = x′ and T ∗x = −x′, so Sx = x′. Then Example 3.4.4(iii) implies graS = {(V x, x) | x ∈ e⊥}. Hence graS−1 = {(x, V x) | x ∈ e⊥}. (3.30) Theorem 3.4.5(iii) implies domS is dense. Furthermore, gra(−S) = {(V x,−x) | x ∈ e⊥}, so gra(−S)−1 = {(x,−V x) | x ∈ e⊥}. (3.31) Since V ∗x(t) = ∫ 1 t x−0 = ∫ 1 t x− ∫ 1 0 x = − ∫ t 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 max- imally 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∗) ∈ graS∗ ⇔ 〈−x, y〉+ 〈x∗, V y〉 = 0, ∀y ∈ e⊥ ⇔ 〈−x+ V ∗x∗, y〉 = 0, ∀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 x(t) = ∫ 1 t −x′ + x(1)e, so that x− V ∗(−x′) ∈ span{e} and (x,−x′) ∈ graS∗. It follows that domS∗ = {x ∈ L2[0, 1] : x is absolutely continuous on [0, 1], x′ ∈ L2[0, 1]}, ranS∗ = L2[0, 1], and S∗x = −x′, ∀x ∈ domS∗. Since 〈S∗x, x〉 = − ∫ 1 0 x′x = − ( 1 2 x(1)2 − 1 2 x(0)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 ⇒ graS = graS∗∗ ⇒ 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 ⇒ T1x = x′ ∈ e⊥. Conversely, if x ∈ L2[0, 1] satisfies 〈e, x〉 = 0, we define t 7→ z(t) = ∫ t0 x, then z is absolutely continuous, z(0) = z(1), T1z = x. Hence ranT1 = e ⊥. 51 3.4. The inverse Volterra operator on L2[0, 1] T1 is skew, because for every x ∈ domT1, we have 〈T1x, x〉 = ∫ 1 0 x′x = 12x(1) 2 − 12x(0)2 = 0. Moreover, T ∗1 = −T1: indeed, as T1 is skew, by Fact 3.2.12, gra(−T1) ⊆ graT ∗1 . To show that T ∗ 1 = −T1, take z ∈ domT ∗1 , ϕ = T ∗1 z. Put Φ(t) =∫ t 0 ϕ. We have ∀y ∈ domT1, ∫ 1 0 y′z = 〈T1y, z〉 = 〈T ∗1 z, y〉 = 〈ϕ, y〉 = ∫ 1 0 yϕ = ∫ 1 0 yΦ′ (3.33) = [Φ(1)y(1) − Φ(0)y(0)] − ∫ 1 0 Φy′. (3.34) Using y = e ∈ domT1 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 ∈ domT1. Since ranT1 = 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 T ∗1 z = ϕ = Φ ′ = −z′. This implies that domT ∗1 ⊆ domT1. Then by Fact 3.2.12, T ∗1 = −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∗) ∈ graA⇔ x ∈ A−1(−x∗). When A is a linear relation, x ∈ (−A)−1(x∗) ⇔ (x,−x∗) ∈ graA ⇔ (−x, x∗) ∈ graA⇔ −x ∈ A−1x∗ ⇔ 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 : domT2 → L2[0, 1] by T2 = V −1◦ . Then (i) T2x = x ′, ∀x ∈ domT2 where domT2 = {x ∈ H : x is absolutely continuous on [0, 1], x′ ∈ H,x(0) = −x(1)}. (3.35) (ii) T ∗2 = −T2, and both T2,−T2 are maximally monotone and skew. Proof. (i): Since V◦x(t) = 12 (∫ t 0 x− ∫ 1 t x ) , 53 3.4. The inverse Volterra operator on L2[0, 1] V◦ is a one-to-one map. Then V −1◦ ( 1 2 ( ∫ t 0 x− ∫ 1 t x) ) = x(t) = ( 1 2 ( ∫ t 0 x− ∫ 1 t x) )′ , which implies T2x = V −1◦ x = x′ for x ∈ ranV◦. As domT2 = ranV◦, by Example 3.4.4(v), ranV◦ can be written as (3.35). (ii): Since domV = domV ∗ = L2[0, 1], V◦ is skew on L2[0, 1], so maxi- mally monotone. Then T2 = V −1◦ is maximally monotone. Since V◦ is skew and domV◦ = L2[0, 1], we have V ∗◦ = −V◦, by Fact 3.4.8, T ∗2 = (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 sum- marize 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: graS $ graT $ gra(−S∗), graS $ graT1 $ gra(−S∗), graS $ graT2 $ gra(−S∗). While S is skew, T, T1, T2 are maximally monotone and T1, T2 are skew. Also, gra(−S) $ gra(T ∗) $ graS∗, gra(−S) $ gra(−T1) $ graS∗, gra(−S) $ gra(−T2) $ graS∗. 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 max- imally monotone skew operator given in Proposition 3.3.3 and Proposi- tion 3.3.7 such that its negative is not maximally monotone. (ii): Even though the skew operator S in Theorem 3.4.6 has domS dense in L2[0, 1], it still admits two distinct maximally monotone and skew exten- sions 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 Theo- 55 3.5. Discussion rem 3.4.6 and Theorem 3.4.9. Now ∀x ∈ domT ∩ domT ∗, we have Tx+ T ∗x = x′ − x′ = 0. Thus T + T ∗ has a proper monotone extension from domT ∩ domT ∗ $ L2[0, 1] to the 0 map on L2[0, 1]. Consequently, T + T ∗ is not maximally monotone. Note that domT ∩ domT ∗ is dense in L2[0, 1] and that domT − domT ∗ 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, T ∗i = −Ti, we conclude that Ti+ T ∗ i = 0 on domTi, a dense subset of L 2[0, 1]; thus, Ti+ T ∗ i fails to be maximally monotone while both Ti, T ∗ i are maximally monotone. 3.5 Discussion The Brezis-Browder Theorem (see Fact 3.2.13) is a very important character- ization 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 Theo- rem 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 Brezis- Browder Theorem in a general Banach space: Let A : X ⇒ X∗ be a monotone linear relation such that graA 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: graG = {(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 mono- tone extensions, instead of using Zorn’s lemma, has been a very active topic [11, 13, 39–41]. In [39], Crouzeix and Ocaña-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 Ocaña-Anaya’s characterizations on monotonicity and maximal monotonic- ity 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 ᵀ +BAᵀ) and (4.4) λk+1, λk+2, . . . , λp be nonpositive eigenvalues of (AB ᵀ +BAᵀ). (4.5) Moreover, let vi be an eigenvector of eigenvalue λi of (AB ᵀ+BAᵀ) satisfying ‖vi‖ = 1, and 〈vi, vj〉 = 0 for 1 ≤ i 6= j ≤ q. It will be convenient to put Idλ = diag(λ1, . . . , λp) = λ1 0 0 · · · 0 0 λ2 0 · · · 0 0 0 λ3 ... ... 0 0 . . . 0 0 0 0 0 λp , V = [v1 v2 . . . vp] . (4.6) 4.1 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 +Mᵀ by V+(M) = span w1, . . . , ws : wi is an eigenvector associated with a positive eigenvalue αi of M +M ᵀ 〈wi, wj〉 = 0 ∀ i 6= j, ‖wi‖ = 1, i, j = 1, . . . , s. V0(M) = span ws+1, . . . , wl : wi is an eigenvector associated with the 0 eigenvalue of M +Mᵀ 〈wi, wj〉 = 0 ∀ i 6= j, ‖wi‖ = 1, i, j = s+ 1, . . . , l. V−(M) = span wl+1, . . . , wn : wi is an eigenvector associated with a negative eigenvalue αi of M +M ᵀ 〈wi, wj〉 = 0 ∀ i 6= j, ‖wi‖ = 1, i, j = l + 1, . . . , n. which is possible since a symmetric matrix always has a complete orthonor- mal 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 ᵀ : 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+Mᵀ) : 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 +Mᵀ)x = 0 and 〈x,Mx〉 = 0. In particular, the orthogonal decomposition holds: Rn = V+(M)⊕V0(M)⊕ V−(M). Proof. (i): Let x ∈ V+(M). Then x = ∑s i=1 liwi for some (l1, . . . , ls) ∈ Rs. Since {w1, · · · , ws} is a set of orthonormal vectors, they are linearly independent so that x 6= 0 ⇔ (l1, . . . , ls) 6= 0. Note that αi > 0 when i = 1, . . . , s and 〈wi, wj〉 = 0 for i 6= j. We have 2〈x,Mx〉 = 〈x, (M +Mᵀ)x〉 = 〈 s∑ i=1 liwi, (M +M ᵀ)( s∑ i=1 liwi)〉 = 〈 s∑ i=1 liwi, s∑ i=1 liαiwi〉 = s∑ i=1 αil 2 i > 0 if x 6= 0. For every x ∈ V+(M) with x = ∑s i=1 liwi, we have (M +Mᵀ)x = s∑ i=1 li(M +M ᵀ)wi = s∑ i=1 αiliwi ∈ V+(M). As αi > 0 for i = 1, . . . , s and {w1, . . . , ws} is an orthonormal basis of V+(M), we conclude that M +M ᵀ : 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 liwi for some (l1, . . . , ll) ∈ Rl. Note that αi ≥ 0 when i = 1, . . . , l and 〈wi, wj〉 = 0 for i 6= j. We have 2〈x,Mx〉 = 〈x, (M +Mᵀ)x〉 = 〈 l∑ i=1 liwi, (M +M ᵀ)( l∑ i=1 liwi)〉 = 〈 l∑ i=1 liwi, l∑ i=1 liαiwi〉 = l∑ i=1 αil 2 i ≥ 0. The proofs for (iii), (iv) are similar to (i), (ii). (v): Obvious. Corollary 4.1.2 The following hold: (i) graT = {(Bᵀu,Aᵀu) | u ∈ V+(BAᵀ)} is strictly monotone. (ii) graT = {(Bᵀu,Aᵀu) | u ∈ V+(BAᵀ) +V0(BAᵀ)} is monotone. (iii) graT = {(Bᵀu,−Aᵀu) | u ∈ V−(BAᵀ)} is strictly monotone. (iv) graT = {(Bᵀu,−Aᵀu) | u ∈ V−(BAᵀ) +V0(BAᵀ))} is monotone. 62 4.1. Auxiliary results on linear relations Proof. As 〈Bᵀu,Aᵀu〉 = 〈u,BAᵀu〉 ∀u ∈ Rn, the result follows from Propo- sition 4.1.1 by letting M = BAᵀ. Lemma 4.1.3 For every subspace S ⊆ Rp, the following hold. dim{(Bᵀu,Aᵀu) | u ∈ S} = dimS. (4.7) dim{(Bᵀu,−Aᵀu) | u ∈ S} = dimS. (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 (ABᵀ +BAᵀ)V = V Idλ . Some basic properties of G are: Lemma 4.1.5 (i) graG = ker(A B). (ii) G0 = kerB,G−1(0) = kerA. (iii) domG = PX(ker(A B)) and ranG = PX∗(ker(A B)). (iv) ran(G+ Id) = PX∗(ker(A−B B)) = PX(ker(A B −A)), and domG = PX(ker(A−B B)), ranG = PX∗(ker(A (B −A)). 63 4.1. Auxiliary results on linear relations (v) dimgraG = 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 2n = dimker(A B) + dim ran Aᵀ Bᵀ = dimgraG+ p. Hence dimgraG = 2n− p. The following result summarizes the monotonicities of G∗ and G. Lemma 4.1.6 The following hold. (i) graG∗ = {(Bᵀu,−Aᵀu) | u ∈ Rp}. (ii) G∗ is monotone⇔ the matrix AᵀB+BᵀA ∈ Rp×p is negative-semidefinite. (iii) Assume G is monotone. Then n ≤ p. Moreover, G is maximally monotone if and only if dimgraG = n = p. Proof. (i): By Lemma 4.1.5(i), we have (x, x∗) ∈ graG∗ ⇔ (x∗,−x) ∈ graG⊥ = ran Aᵀ Bᵀ = {(Aᵀu,Bᵀu) | u ∈ Rp}. Thus graG∗ = {(Bᵀu,−Aᵀu) | u ∈ Rp}. 64 4.1. Auxiliary results on linear relations (ii): Since graG∗ is a linear subspace, by (i), G∗ is monotone⇔ 〈Bᵀu,−Aᵀu〉 ≥ 0, ∀u ∈ Rp ⇔ 〈u,−BAᵀu〉 ≥ 0, ∀u ∈ Rp ⇔ 〈u,BAᵀu〉 ≤ 0, ∀u ∈ Rp ⇔ 〈u, (AᵀB +BᵀA)u〉 ≤ 0, ∀u ∈ Rp ⇔ (AᵀB +BᵀA) is negative semidefinite. (iii): By Fact 3.2.6 and Lemma 4.1.5(v), 2n− p = dimgraG ≤ n⇒ n ≤ p. By Fact 3.2.6 and Lemma 4.1.5(v) again, G is maximally monotone ⇔ 2n − p = dimgraG = n⇔ dimgraG = p = n. 4.1.1 One linear relation: two equivalent formulations The linear relation G given by (4.1)–(4.3): graG = {(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 x∗ = h1y1 + · · ·+ h2n−py2n−p = C D y 65 4.2. Explicit maximally monotone extensions of monotone linear relations where y ∈ R2n−p and C D = (h1, . . . , h2n−p) with C,D being n× (2n − p) matrices. Therefore, graG = Cy Dy ∣∣∣∣∣∣∣ y ∈ R2n−p = ran C 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 exten- sions of G by using V+(AB ᵀ) or Vg. A characterization of all maximally monotone extensions of G is also given. We also provide a new proof for Crouzeix and Ocaña-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 G̃ and Ĝ be defined by gra G̃ = {(x, x∗) |MᵀAx+MᵀBx∗ = 0} gra Ĝ = {(Bᵀu,−Aᵀu) | u ∈ ranM}. 66 4.2. Explicit maximally monotone extensions of monotone linear relations Then (G̃)∗ = Ĝ. Proof. Let (y, y∗) ∈ Rn ×Rn. Then we have (y, y∗) ∈ gra(G̃)∗ ⇔ (y∗,−y) ∈ (gra G̃)⊥ = (ker ( MᵀA MᵀB ))⊥ = ran AᵀM BᵀM ⇔ (y, y∗) ∈ gra Ĝ. Hence (G̃)∗ = Ĝ. Lemma 4.2.2 Define linear relations G̃ and Ĝ by gra G̃ = {(x, x∗) | VgAx+ VgBx∗ = 0} gra Ĝ = {(Bᵀu,−Aᵀu) | u ∈ V−(BAᵀ) +V0(BAᵀ)}, where Vg is (p − k)× p matrix defined by Vg = vᵀk+1 vᵀk+2 ... vᵀp . Then (i) Ĝ is monotone. (ii) (Ĝ)∗ = G̃. 67 4.2. Explicit maximally monotone extensions of monotone linear relations (iii) gra G̃ = graG+ Bᵀ Aᵀ u ∣∣∣∣∣∣∣u ∈ V+(BAᵀ) . Proof. (i): Apply Corollary 4.1.2(iv). (ii): Notations are as in (4.6). Define the p× p matrix N by N = 0 0 0 Id in which Id ∈ R(p−k)×(p−k). Then we have NᵀV ᵀ = ( (v1 · · · vk V ᵀg ) 0 0 0 Id )ᵀ = 0 Vg . (4.11) Then we have VgAx+ VgBx ∗ = 0⇔ 0 VgAx+ VgBx ∗ = 0 ⇔ NᵀV ᵀAx+NᵀV ᵀBx∗ = 0, ∀(x, x∗) ∈ Rn × Rn. Hence gra G̃ = {(x, x∗) | NᵀV ᵀAx+NᵀV ᵀBx∗ = 0}. Thus by Lemma 4.2.1 with M = V N , gra(G̃)∗ = {(Bᵀu,−Aᵀu) | u ∈ ranV N = ran ( 0 V ᵀg ) 68 4.2. Explicit maximally monotone extensions of monotone linear relations = V−(BAᵀ) +V0(BAᵀ)} = gra Ĝ. Hence (Ĝ)∗ = (G̃)∗∗ = G̃. (iii): Let J be defined by gra J = graG+ Bᵀ Aᵀ u ∣∣∣∣∣∣∣u ∈ V+(BAᵀ) . Then we have (gra J)⊥ = (graG)⊥ ∩ Bᵀ Aᵀ u ∣∣∣∣∣∣∣ u ∈ V+(BAᵀ) ⊥ . By Lemma 4.1.5(i), graG⊥ = Aᵀ Bᵀ w ∣∣∣∣∣∣∣w ∈ Rp Then Aᵀ Bᵀ w ∈ Bᵀ Aᵀ u ∣∣∣∣∣∣∣ u ∈ V+(BAᵀ) ⊥ if and only if 〈(Aᵀw,Bᵀw), (Bᵀu,Aᵀu)〉 = 0 ∀ u ∈ V+(BAᵀ), 69 4.2. Explicit maximally monotone extensions of monotone linear relations that is, 〈Aᵀw,Bᵀu〉+ 〈Bᵀw,Aᵀu〉 = 〈w, (ABᵀ +BAᵀ)u〉 = 0 ∀u ∈ V+(ABᵀ). (4.12) Because ABᵀ+BAᵀ : V+(AB ᵀ) 7→ V+(ABᵀ) is onto by Proposition 4.1.1(i), we obtain that (4.12) holds if and only if w ∈ V−(ABᵀ)+V0(ABᵀ). Hence (gra J)⊥ = {(Aᵀw,Bᵀw) | w ∈ V−(BAᵀ) +V0(BAᵀ)}, from which gra J∗ = gra Ĝ. Then by (i), gra G̃ = gra(Ĝ)∗ = gra J∗∗ = gra J. We are ready to apply the Brezis-Browder Theorem, namely Fact 3.2.13, to improve Crouzeix and Ocaña-Anaya’s characterizations of monotonicity and maximal monotonicity of G and provide a different proof. Theorem 4.2.3 Let Ĝ, G̃ be defined in Lemma 4.2.2. The following are equivalent: (i) G is monotone; (ii) G̃ is monotone; (iii) G̃ is maximally monotone; (iv) Ĝ is maximally monotone; 70 4.2. Explicit maximally monotone extensions of monotone linear relations (v) dimV+(BA ᵀ) = p − n, equivalently, ABᵀ + BAᵀ 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 G̃ = (Ĝ)∗ and Ĝ 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 G̃ 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, Ĝ is maximally monotone, so that dim(gra Ĝ) = 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 Ĝ) = p − k = n by Lemma 4.1.3, so that Ĝ is maximally monotone by Fact 3.2.6(i)(ii). By Lemma 4.2.2(ii) and Fact 3.2.13, G̃ is monotone, which implies that G is monotone. Corollary 4.2.4 Assume that G is monotone. Then gra G̃ = graG+ Bᵀ Aᵀ u ∣∣∣∣∣∣∣ u ∈ V+(BAᵀ〉 = {(x, x∗) | VgAx+ VgBx∗ = 0} 71 4.2. Explicit maximally monotone extensions of monotone linear relations is a maximally monotone extension of G, where Vg = vᵀp−n+1 vᵀp−n+2 ... vᵀp . 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 ex- tensions 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 Ocaña-Anaya. Remark 4.2.5 (i). Crouzeix and Ocaña-Anaya [39] defines the union of monotone extension of G as S = graG+ Bᵀ Aᵀ u ∣∣∣∣∣∣∣u ∈ K , where K = {u ∈ Rn | 〈u, (ABᵀ +BAᵀ)u〉 ≥ 0}. Although this is the set mono- tonically related to G, it is not monotone in general as long as (ABᵀ+BAᵀ) has both positive eigenvalues and negative eigenvalues. Indeed, let (α1, u1) and (α2, u2) be eigen-pairs of (AB ᵀ + BAᵀ) with α1 > 0 and α2 < 0. We have 〈u1, (ABᵀ+BAᵀ)u1〉 = α1‖u1‖2 > 0, 〈u2, (ABᵀ+BAᵀ)u2〉 = α2‖u2‖2 < 0. 72 4.2. Explicit maximally monotone extensions of monotone linear relations Choose > 0 sufficiently small so that 〈u1 + u2, (ABᵀ +BAᵀ)(u1 + u2)〉 > 0. Then Bᵀ Aᵀ u1, Bᵀ Aᵀ (u1 + u2) ∈ S. However, Bᵀ Aᵀ (u1 + u2)− Bᵀ Aᵀ u1 = Bᵀ Aᵀ u2 has 〈Bᵀu2, Aᵀu2〉 = 2〈u2, BAᵀu2〉 = 2 〈u2, (AB ᵀ +BAᵀ)u2〉 2 < 0. Therefore S is not monotone. By using V+(BA ᵀ) ⊆ K, we have obtained a maximally monotone extension of G . (ii). Crouzeix and Ocaña-Anaya [39] find a maximally monotone linear subspace extension of G algorithmically by using ũk ∈ gra G̃k \ graGk and constructing graGk+1 = graGk + Rũk where ũk = Bᵀk Aᵀk uk, 〈uk, (AkBᵀk +BkAᵀk)uk〉 ≥ 0. This recursion is done until dimgraGk = n. In particular, each uk may be chosen as an eigenvector associated with a positive eigenvalue of AkB ᵀ k + BkA ᵀ k, 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 Ocaña-Anaya [39]. Theorem 4.2.6 Let Ĝ, G̃ 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 ABᵀ +BAᵀ is negative semidefinite. (iv) p = n and Ĝ 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 (ABᵀ+BAᵀ) is negative semidefinite. Then k = 0 and G̃ = G. It follows that dim(gra Ĝ) = p − k = n by Lemma 4.1.3, so that Ĝ is maximally monotone by Corollary 4.1.2(iv) and Fact 3.2.6(i)(ii). Since ( Ĝ )∗ = G̃ by Lemma 4.2.2(ii), Fact 3.2.13 gives that G̃ = G is maximally monotone. (iii)⇒(iv): Assume that p = n and (ABᵀ+BAᵀ) is negative semidefinite. We have k = 0 and dim(gra Ĝ) = 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 Ĝ is maximally monotone and p = n. We have dim(gra Ĝ) = p − k = n − k = n so that k = 0. Hence (ABᵀ + BAᵀ) 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 G̃ is a maximally monotone extension of G if and only if there exists N ∈ Rp×p with rank of n such that Nᵀ IdλN is negative semidefinite and gra G̃ = {(x, x∗) | NᵀV ᵀAx+NᵀV ᵀBx∗ = 0}. (4.13) Proof. “⇒”: By Lemma 4.1.6(i), we have graG∗ = {(Bᵀu,−Aᵀu) | u ∈ Rp}. (4.14) Since graG ⊆ gra G̃ and thus gra(G̃)∗ is a subspace of graG∗. Thus by (4.14), there exists a subspace F of Rp such that gra(G̃)∗ = {(Bᵀu,−Aᵀu) | u ∈ F}. (4.15) By Fact 3.2.13, Fact 3.2.6 and Lemma 4.1.3, we have dimF = 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 ranV N = F and gra(G̃)∗ = {(BᵀV Ny,−AᵀV Ny) | y ∈ Rp}. (4.17) As G̃ is maximally monotone, (G̃)∗ is maximally monotone by Fact 3.2.13, so NᵀV ᵀ(BAᵀ +ABᵀ)V N is negative semidefinite. Using Fact 4.1.4, we have Nᵀ IdλN = N ᵀV ᵀV IdλN = N ᵀV ᵀ(ABᵀ +BAᵀ)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(G̃)∗ = {(BᵀV Nu,−AᵀV Nu) | u ∈ Rp}. (4.19) Observe that (G̃)∗ is monotone because NᵀV ᵀ(ABᵀ+BAᵀ)V N = Nᵀ IdλN 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(G̃)∗ = n. Therefore (G̃)∗ is maximally monotone by Fact 3.2.6. Applying Fact 3.2.13 for T = (G̃)∗ yields that G̃ = (G̃)∗∗ is maximally monotone. From the above proof, we see that to find a maximally monotone exten- sion of G one essentially need to find subspace F ⊆ Rp such that dimF = n and ABᵀ+BAᵀ is negative semidefinite on F . If F = ranM and M ∈ Rp×p 76 4.2. Explicit maximally monotone extensions of monotone linear relations with rankM = n, one can let N = V ᵀM . The maximally monotone linear subspace extension of G is G̃ = {(x, x∗) |MᵀAx+MᵀBx∗ = 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 G̃ is a maximally monotone extension of G if and only if there exists M ∈ Rp×p with rank of n such that Mᵀ(ABᵀ +BAᵀ)M is negative semidefinite and gra G̃ = {(x, x∗) |MᵀAx+MᵀBx∗ = 0}. (4.20) Note that G may have different representations in terms of A,B. The maximally monotone extension of G̃ given in Theorem 4.2.7 and Corol- lary 4.2.4 relies on A,B matrices and N . This might lead to different max- imally 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 G̃ = {(x, x∗) ∈ Rn × Rn | Ãx+ B̃x∗ = 0} ⊇ graG where Ã, B̃ ∈ 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 graG = {(x, x∗) ∈ Rn × Rn | V ᵀAx+ V ᵀBx∗ = 0}. As G̃ is maximally monotone, the n × 2n matrix has rank(Ã, B̃) = n and the matrix ÃB̃ᵀ + B̃Ãᵀ ∈ Rn×n is negative semidefinite. Since gra G̃ ⊇ graG, we have ran Ãᵀ B̃ᵀ = (gra G̃)⊥ ⊆ (graG)⊥ = ran (V ᵀA)ᵀ (V ᵀB)ᵀ . Therefore, there exists a p× n matrix N with rankN = n such that Ãᵀ B̃ᵀ = (V ᵀA)ᵀ (V ᵀB)ᵀ N = (V ᵀA)ᵀN (V ᵀB)ᵀN from which Ã = NᵀV ᵀA, B̃ = NᵀV ᵀB. Then the n× n matrix ÃB̃ᵀ + B̃Ãᵀ = NᵀV ᵀA(NᵀV ᵀB)ᵀ +NᵀV ᵀB(NᵀV ᵀA)ᵀ (4.21) = NᵀV ᵀ(ABᵀ +BAᵀ)V N (4.22) = Nᵀ IdλN. (4.23) Therefore, all maximally monotone linear subspace extensions of G can be 78 4.3. Minty parameterizations obtained by using gra G̃ = {(x, x∗) ∈ Rn × Rn | NᵀV ᵀAx+NᵀV ᵀBx∗ = 0} in which the p× n matrix N satisfies rankN = n and Nᵀ IdλN 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 ‖y‖2 − ‖y∗‖2 ≥ 0, whenever (4.24) (A+B)y + (B −A)y∗ = 0. (4.25) 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 P = 0 Id 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∗), P x x∗ 〉 ≥ 0, whenever Ax+Bx∗ = 0. Define the orthogonal matrix Q = 1√ 2 Id − Id Id Id and put x x∗ = Q y y∗ . Then G is monotone if and only if ‖y‖2 − ‖y∗‖2 ≥ 0, whenever (4.26) (A+B)y + (B −A)y∗ = 0. (4.27) If (B − A) does not have full column rank, then there exists y∗ 6= 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 mono- tone operator. Then (x, x∗) ∈ graG if and only if x = 1 2 [Id+(B −A)†(B +A)]y (4.28) x∗ = 1 2 [Id−(B −A)†(B +A)]y (4.29) 80 4.3. Minty parameterizations for y = x+ x∗ ∈ ran(Id+G). Here the Moore-Penrose inverse (B − A)† = [(B−A)ᵀ(B−A)]−1(B−A)ᵀ. In particular, when G is maximally monotone, we have graG = {((B −A)−1By,−(B −A)−1Ay) | y ∈ Rn}. Proof. As (B − A) is full column rank, (B − A)ᵀ(B − A) is invertible. It follows from (4.25) that (B−A)ᵀ(A+B)y+(B−A)ᵀ(B−A)y∗ = 0 so that y∗ = −((B −A)ᵀ(B −A))−1(B −A)ᵀ(A+B)y = −(B −A)†(A+B)y. Then x = 1√ 2 (y − y∗) = 1√ 2 [Id+(B −A)†(B +A)]y x∗ = 1√ 2 (y + y∗) = 1√ 2 [Id−(B −A)†(B +A)]y where y = x+x ∗√ 2 with (x, x∗) ∈ graG. Since ran(Id+G) is a subspace, we have x = 1 2 [Id+(B −A)†(B +A)]ỹ x∗ = 1 2 [Id−(B −A)†(B +A)]ỹ with ỹ = 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 x = 1 2 (B −A)−1[B −A+ (B +A)]y = (B −A)−1By (4.30) x∗ = 1 2 (B −A)−1[(B −A)− (B +A)]y = −(B −A)−1Ay (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 7→ ((G+ Id)−1, Id−(G+ Id)−1)(z) is bijective and linear from ran(G+Id) to graG, therefore dim(ran(G+Id)) = dim(graG). Corollary 4.3.4 Let G be a monotone operator. Then G̃ defined in Corol- lary 4.2.4, the maximally monotone extension of G, has its Minty parametriza- tion given by gra G̃ = {((VgB − VgA)−1VgBy,−(VgB − VgA)−1VgAy) | 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(VgA VgB) = 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, domG = (B −A)−1(ranB), ranG = (B −A)−1(ranA). Recall that T : Rn → Rn is firmly nonexpansive if ‖Tx− Ty‖2 ≤ 〈Tx− Ty, x− y〉 ∀ x, y ∈ domT. In terms of matrices, we have Corollary 4.3.6 Suppose that p = n, ABᵀ +BAᵀ is negative semidefinite. Then (B −A)−1B and −(B −A)−1A are firmly nonexpansive. Proof. By Theorem 4.2.6, G is maximally monotone. Theorem 4.3.2 gives that (B −A)−1B = (Id+G)−1, −(B −A)−1A = (Id+G−1)−1. Being resolvents of monotone operators G,G−1, they are firmly nonexpan- sive, 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 domG or ranG 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 rela- tion. Then (i) T1 = T +NdomT , i.e., x 7→ T1x = Tx+ (domT )⊥ if x ∈ domT ∅ otherwise is maximally monotone. In particular, domT1 = domT . (ii) T2 = (T −1 + NranT )−1 is a maximally monotone extension of T and ranT2 = ranT . Proof. (i): Since 0 ∈ T0 ⊆ (dom T )⊥ by [15, Proposition 2.2(i)], we have T10 = T0+(domT ) ⊥ = (domT )⊥ so that domT1 = domT = (T10)⊥. Hence T1 is maximally monotone by Fact 3.2.6. (ii): Apply (i) to T−1 to see that T−1 + NranT is a maximally mono- tone extension of T−1 with dom(T−1 +NranT ) = ranT . Therefore, T2 is a maximally monotone extension of T with ranT2 = ranT . Define linear relations Ei : Rn ⇒ Rn (i = 1, 2) by graE1 = Cy Dy + 0 (ranC)⊥ ∣∣∣∣∣∣∣ y ∈ R2n−p , (4.32) 84 4.4. Maximally monotone extensions with the same domain or the same range graE2 = Cy Dy + (ranD)⊥ 0 ∣∣∣∣∣∣∣ y ∈ R2n−p . (4.33) Theorem 4.4.2 (i) E1 is a maximally monotone extension of G with domE1 = domG. Moreover, graE1 = ran C D + 0 (ranC)⊥ = ran C D + 0 kerCᵀ . (4.34) (ii) E2 is a maximally monotone extension of G with ranE2 = ranG. Moreover, graE2 = ran C D + (ranD)⊥ 0 = ran C D + kerDᵀ 0 . (4.35) Proof. (i): Note that domG = ranC. The maximal monotonicity fol- lows from Proposition 4.4.1. (4.34) follows from (4.32) and the fact that (ranC)⊥ = kerCᵀ [59, page 405]. (ii): Apply (i) to G−1, i.e., graG−1 = Dy Cy ∣∣∣∣∣∣∣ y ∈ R2n−p (4.36) and followed by taking the set-valued inverse. Apparently, both extensions E1, E2 rely on graG, domG, ranG, 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(E∗1) = {(Bᵀu,−Aᵀu) | Bᵀu ∈ domG,u ∈ Rp}. (4.37) gra(E∗2) = {Bᵀu,−Aᵀu) | Aᵀu ∈ ranG,u ∈ Rp}. (4.38) According to Fact 3.2.13, E∗i is maximally monotone and dimE ∗ i = n. This implies that dim{u ∈ Rp | Bᵀu ∈ domG} = n, dim{u ∈ Rp | Aᵀu ∈ ranG} = n. Let Mi ∈ Rp×p with rankMi = n and {u ∈ Rp | Bᵀu ∈ domG} = ranM1, (4.39) {u ∈ Rp | Aᵀu ∈ ranG} = ranM2. (4.40) Corollary 4.2.8 shows that graEi = {(x, x∗) |Mᵀi Ax+Mᵀi 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 maxi- mally 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 con- sidering three examples. In particular, they show that maximally monotone extensions G̃ 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 graG. Example 4.5.1 Consider graG = (x, x∗) ∈ Rn × Rn ∣∣∣∣∣∣∣ Id 0 x+ 0 C x∗ = 0 where C ∈ Rn×n is symmetric and positive definite, and Id ∈ Rn×n . Clearly, graG = 0 0 . We have (i) For every α ∈ [−1, 1] , G̃α defined by gra G̃α = {(0,Rn)} , if α = 1;{ (x, 1+α1−αC −1x) | x ∈ Rn } , otherwise is a maximally monotone linear extension of G. 87 4.5. Examples (ii) E1 = G̃1 and E2 = G̃−1. Proof. (i): To find G̃α, we need eigenvectors of A = Id 0 (0 Cᵀ) + 0 C (Id 0) = 0 C 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 6= j. As such, the matrix A has 2n eigen-pairs, namely λi, wi wi and −λi, wi −wi with i = 1, . . . , n. Put W = (w1 · · · wn) ∈ Rn×n and write V = W W W −W . Then W ᵀCW = D = diag(λ1, λ2, . . . , λn). 88 4.5. Examples In Theorem 4.2.7, take Nα = 0 α Id 0 Id ∈ R2n×2n where Id ∈ Rn×n. We have rankNα = n, Nᵀα IdλNα = 0 0 0 (α2 − 1)W ᵀCW = 0 0 0 (α2 − 1)D being negative semidefinite, and V Nα = 0 (1 + α)W 0 (α− 1)W . Then by Theorem 4.2.7, we have a maximally monotone linear extension G̃α given by gra G̃α = (x, x∗) ∈ Rn × Rn ∣∣∣∣∣∣∣ 0 (1 + α)W ᵀx+ (α− 1)W ᵀCx∗ = 0 = { (x, x∗) ∈ Rn × Rn | (1 + α)x+ (α− 1)Cx∗ = 0} = {(0,Rn)} , if α = 1;{ (x, 1+α1−αC −1x) | x ∈ Rn } , otherwise. 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 graG = (x, x ∗) ∈ R2 × R2 ∣∣∣∣∣∣∣∣∣∣ −1 0 0 0 0 −1 x1 x2 + 1 0 0 1 0 1 x∗1 x∗2 = 0 . Then (i) the linear operators G̃i : R2 ⇒ R2 for i = 1, 2 given by G̃1 = 1 0 0 −1+ √ 2 2−√2 , G̃2 = 1 25 0 √ 2 10 are two maximally monotone extensions of G. (ii) E1(x1, 0) = (x1,R) ∀x1 ∈ R. (iii) E2(x1, y) = (x1, 0) ∀x1, y ∈ R. Proof. We have graG = x1 0 x1 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1 ∈ R is monotone. Since dimG = 1, G is not maximally monotone by Fact 3.2.6. 90 4.5. Examples The matrix ABᵀ +BAᵀ = −2 0 0 0 0 −1 0 −1 −2 has a positive eigenvalue −1 +√2 with an eigenvector u = 0 1 1−√2 so that Bᵀ Aᵀ u = 0 2−√2 0 −1 +√2 . Then by Corollary 4.2.4, gra G̃1 = x1 0 x1 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1 ∈ R + 0 2−√2 0 −1 +√2 x2 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x2 ∈ R = x1 (2−√2)x2 x1 (−1 +√2)x2 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1, x2 ∈ R . Therefore, G̃1 = 1 0 0 −1+ √ 2 2−√2 is a maximally monotone extension of G. 91 4.5. Examples Now we have Idλ = −1 +√2 0 0 0 −1−√2 0 0 0 −2 , V = 0 0 1 − 1−1+√2 − 1 −1−√2 0 1 1 0 . (4.41) Take N = 0 −1 1 0 2 −1 0 1 1 . (4.42) We have rankN = 2 and Nᵀ IdλN = 0 0 0 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)ᵀAx+ (V N)ᵀBx∗ = 0, 92 4.5. Examples and get gra G̃2 = span 1 0 1 0 , −2√2 5 √ 2 0 1 . Thus G̃2 = 1 −2√2 0 5 √ 2 −1 = 1 25 0 √ 2 10 is another maximally monotone extension of G. On the other hand, graE1 = x1 0 x1 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1 ∈ R + 0 0 0 R = x1 0 x1 R ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1 ∈ R gives E1(x1, 0) = (x1,R) ∀x1 ∈ R. We have graE2 = x1 R x1 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1 ∈ R , which gives E2(x1, y) = (x1, 0) ∀x1, y ∈ R. 93 4.5. Examples In [11], the authors use autoconjugates to find maximally monotone ex- tensions 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 graG = {(x, x∗) ∈ R2 × R2 | Ax + Bx∗ = 0} where A = 1 1 2 0 3 1 , B = 1 5 1 7 0 2 , thus (A B) = 1 1 1 5 2 0 1 7 3 1 0 2 . (4.44) Then the linear operators G̃i : Rn ⇒ Rn for i = 1, 2 given by G̃1 = −117+17 √ 201 2(−1+√201) −107+7√201 2(−1+√201) − −23+3 √ 201 2(−1+√201) − −21+√201 2(−1+√201) , G̃2 = 334 − √ 201 6 13 4 − √ 201 6 −2920 + √ 201 30 − 920 + √ 201 30 are two maximally monotone linear extensions of G. Moreover, graE1 = −1 1 −5 1 x1 + 0 0 1 1 x2 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1, x2 ∈ R 94 4.5. Examples and graE2 = −1 1 −5 1 x1 + 1 5 0 0 x2 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1, x2 ∈ R . Proof. We have rank(A B) = 3 and Idλ = 13 + √ 201 0 0 0 −6 0 0 0 13−√201 , V = 20 1+ √ 201 0 20 1−√201 1 −1 1 1 1 1 , (4.45) and Vg = 0 −1 1 20 1−√201 1 1 . (4.46) Clearly, here p = 3, n = 2 and ABᵀ + BAᵀ has exactly p − n = 3 − 2 = 1 positive eigenvalue. By Theorem 4.2.3(i)(v), G is monotone. Since ABᵀ +BAᵀ 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 VgAx+ VgBx ∗ = 0 and obtain G̃1 defined by gra G̃1 = span − −21+ √ 201 2(−1+√201) −23+3√201 2(−1+√201) 1 0 , −−107+7 √ 201 2(−1+√201) −117+17√201 2(−1+√201) 0 1 . By Corollary 4.2.4, G̃1 is a maximally monotone linear subspace extension of G. Then G̃1 = − −21+ √ 201 2(−1+√201) − −107+7√201 2(−1+√201) −23+3√201 2(−1+√201) −117+17√201 2(−1+√201) −1 = −117+17 √ 201 2(−1+√201) −107+7√201 2(−1+√201) − −23+3 √ 201 2(−1+√201) − −21+√201 2(−1+√201) . Let N be defined by N = 0 0 15 0 1 0 0 0 1 . (4.47) Then rankN = 2 and Nᵀ IdλN = 0 0 0 0 −6 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)ᵀAx+(V N)ᵀBx∗ = 0. By Theorem 4.2.7, 96 4.5. Examples we get a maximally monotone linear extension of G, G̃2, defined by G̃2 = − 920 + √ 201 30 −134 + √ 201 6 29 20 − √ 201 30 33 4 − √ 201 6 −1 = 334 − √ 201 6 13 4 − √ 201 6 −2920 + √ 201 30 − 920 + √ 201 30 . To find E1 and E2, using the LinearSolve command in Maple, we get graG = ran C D , where C = −1 1 , D = −5 1 . It follows from Theorem 4.4.2 that graE1 = −1 1 −5 1 x1 + 0 0 1 1 x2 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1, x2 ∈ R and graE2 = −1 1 −5 1 x1 + 1 5 0 0 x2 ∣∣∣∣∣∣∣∣∣∣∣∣∣ x1, x2 ∈ R . 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 7→ 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 domA ∩ int dom B 6= ∅ 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 chap- ter, we prove the maximal monotonicity of A + B provided that domA ∩ int domB 6= ∅, A+NdomB is of type (FPV), and domA∩domB ⊆ domB. 99 5.1. Basic properties We also show the maximal monotonicity of A + B when A is a maxi- mally monotone linear relation and B is a subdifferential operator satisfying domA ∩ int domB 6= ∅. 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 domA 6= ∅. Assume that x ∈ int domA. 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) ∩ domA. 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 6= ∅. Then f is continuous on int dom f and ∂f(x) 6= ∅ 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∗) 7→ sup (a,a∗)∈graA (〈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∗) ∈ graA. Fact 5.1.6 (Fitzpatrick) (See [45, Theorem 3.4].) Let A : X ⇒ X∗ be monotone. Then conv domA ⊆ PX(domFA). 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(domFA)− PX(domFB)] 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 domA 6= ∅. Then int domA = int [PX domFA], domA = PX [domFA], and domA 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 := sup a∈X,a6=x λ− f(a) ‖x− a‖ . 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 6= x and y∗ 6= x∗ such that ∣∣∣∣ ‖x− y‖‖x∗ − y∗‖ − α ∣∣∣∣ < ε (5.5) and ∣∣∣∣ 〈x− y, x∗ − y∗〉‖x− y‖ · ‖x∗ − y∗‖ + 1 ∣∣∣∣ < ε. (5.6) Fact 5.1.11 (Simons) (See [74, Corollary 28.2].) Let A : X ⇒ X∗ be max- imally monotone. Then span(PX domFA) = span [domA]. (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, Corol- lary 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 domA = conv domA = PX domFA. 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 domA ∩ intC 6= ∅, 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 domA∩ intC 6= ∅. 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 domA ∩ intC 6= ∅. Then A+NC is of type (FPV ). Proof. By Fact 5.1.16, A+NC is maximally monotone. LetD be a nonempty closed convex subset of X, and suppose that dom(A+NC)∩ intD 6= ∅. Let x1 ∈ domA ∩ intC and x2 ∈ dom(A + NC) ∩ intD. 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) + 12δUX ⊆ D. Clearly, x2 + λ(x1 − x2) + λδUX ⊆ C. Thus x2 + λ(x1 − x2) + λδ2 UX ⊆ C ∩D. Since domA is convex, x2 + λ(x1 − x2) ∈ domA and x2 + λ(x1 − x2) ∈ domA ∩ 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 (FPV ). Corollary 5.1.18 Let A : X ⇒ X∗ be a maximally monotone linear re- lation, let C be a nonempty closed convex subset of X, and suppose that domA ∩ intC 6= ∅. Then A+NC is of type (FPV ). 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 intC 6= ∅. Let c0 ∈ intC and suppose that z ∈ X r C. Then there exists λ ∈ ]0, 1[ such that λc0 + (1− λ)z ∈ bdryC. 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 ∈ bdryC. Assume to the contrary that λc0+(1−λ)z ∈ intC. 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 ∈ bdryC. Since c0 /∈ bdryC, we also have λ < 1. The proof of the next result follows closely the proof of [74, Theo- rem 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 domA ∩ int dom ∂f 6= ∅, (z, z∗) ∈ X × X∗ is monotonically related to gra(A+ ∂f) and z ∈ domA. Then z ∈ dom ∂f . Proof. Let c0 ∈ X and y∗ ∈ X∗ be such that c0 ∈ domA ∩ int dom ∂f and (z, y∗) ∈ graA. (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 7→ 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 Kλ := sup g(x)<λ λ− g(x) ‖x‖ . (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∗) ∈ graA. 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 domA ∩ int domB 6= ∅. Assume that A +NdomB is maximally monotone of type (FPV), and domA ∩ domB ⊆ domB. Then PX [domFA+B ] = 107 5.2. Maximality of the sum of a (FPV) operator and a full domain operator domA ∩ domB. Proof. By Fact 5.1.6, domA ∩ domB = dom(A+B) ⊆ PX [domFA+B ]. It suffices to show that PX [domFA+B ] ⊆ domA ∩ domB. (5.18) After translating the graphs if necessary, we can and do assume that 0 ∈ domA ∩ int domB and (0, 0) ∈ graB. To show (5.18), we take z ∈ PX [domFA+B ] and we assume to the con- trary that z /∈ domA ∩ domB. (5.19) Thus α = d(z,domA ∩ domB) > 0. Now take y∗0 ∈ X∗ such that ‖y∗0‖ = 1 and 〈z, y∗0〉 ≥ 23‖z‖. (5.20) Set Un = [0, z] + α 4nUX , ∀n ∈ N. (5.21) Since 0 ∈ NdomB(x),∀x ∈ domB, graB ⊆ gra(B + NdomB). Since B is maximally monotone and B + NdomB is a monotone extension of B, we must have B = B +NdomB. Thus A+B = A+NdomB +B. (5.22) 108 5.2. Maximality of the sum of a (FPV) operator and a full domain operator Since domA ∩ domB ⊆ domB by assumption, we obtain domA ∩ domB ⊆ dom(A+NdomB) = domA ∩ domB ⊆ domA ∩ domB. Hence domA ∩ domB = dom(A+NdomB). (5.23) By (5.19) and (5.23), z /∈ dom(A + NdomB) and thus (z, ny∗0) /∈ gra(A + NdomB),∀n ∈ N. For every n ∈ N, since z ∈ Un and since A+NdomB is of type (FPV) by assumption, we deduce the existence of (zn, z ∗ n) ∈ gra(A + NdomB) such that zn ∈ Un and 〈z − zn, z∗n〉 > n〈z − zn, y∗0〉, ∀n ∈ N. (5.24) Hence, using (5.21), there exists λn ∈ [0, 1] such that ‖z − zn − λnz‖ = ‖zn − (1− λn)z‖ < 14α, ∀n ∈ N. (5.25) By the triangle inequality, we have ‖z − zn‖ < λn‖z‖+ 14α for every n ∈ N. From the definition of α and (5.23), it follows that α ≤ ‖z − zn‖ and hence that α < λn‖z‖+ 14α. 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 − λnz, y∗0〉 ≥ −‖zn − (1− λn)z‖ > −14α, ∀n ∈ N. (5.27) By (5.27), (5.20) and (5.26), 〈z − zn, y∗0〉 > λn〈z, y∗0〉 − 14α > 23 34α− 14α = 14α, ∀n ∈ N. (5.28) Then, by (5.24) and (5.28), 〈z − zn, z∗n〉 > 14nα, ∀n ∈ N. (5.29) By (5.21), there exist tn ∈ [0, 1] and bn ∈ α4nUX such that zn = tnz + 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 ∈ domA ∩ domB; 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+N domB +B(z, z ∗) ≥ sup {n∈N,y∗∈X∗} [〈zn, z∗〉+ 〈z − zn, z∗n〉+ 〈z − zn, y∗〉 − ιgraB(zn, y∗)] ≥ sup {n∈N,y∗∈X∗} [〈zn, z∗〉+ 14nα+ 〈z − zn, y∗〉 − ιgraB(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 domB), there exist N ∈ N and K > 0 such that Bzn 6= ∅ and sup y∗∈Bzn ‖y∗‖ ≤ K, ∀n ≥ N. (5.33) Then, by (5.31), FA+B(z, z ∗) ≥ sup {n≥N,y∗∈X∗} [〈zn, z∗〉+ 14nα+ 〈z − zn, y∗〉 − ιgraB(zn, y∗)] ≥ sup {n≥N,y∗∈Bzn} [−‖zn‖ · ‖z∗‖+ 14nα− ‖z − zn‖ · ‖y∗‖] ≥ sup {n≥N} [−‖zn‖ · ‖z∗‖+ 14nα−K‖z − zn‖] (by (5.33)) =∞ (by (5.30)). Thus (5.32) holds. Case 2 : β 6= 0. Take v∗n ∈ Bzn. We consider two subcases. Subcase 2.1 : (v∗n)n∈N is bounded. By (5.31), FA+B(z, z ∗) ≥ sup {n∈N} [〈zn, z∗〉+ 14nα+ 〈z − zn, v∗n〉] 111 5.2. Maximality of the sum of a (FPV) operator and a full domain operator ≥ sup {n∈N} [−‖zn‖ · ‖z∗‖+ 14nα− ‖z − zn‖ · ‖v∗n‖] =∞ (by (5.30) and the boundedness of (v∗n)n∈N). Hence (5.32) holds. Subcase 2.2 : (v∗n)n∈N is unbounded. We first show lim sup n→∞ 〈z − zn, v∗n〉 ≥ 0. (5.34) Since (v∗n)n∈N is unbounded and after passing to a subsequence if necessary, we assume that ‖v∗n‖ 6= 0,∀n ∈ N and that ‖v∗n‖ → +∞. By 0 ∈ int domB and Fact 5.1.3, there exist δ > 0 and M > 0 such that By 6= ∅ and sup y∗∈By ‖y∗‖ ≤M, ∀y ∈ δBX . (5.35) Then we have 〈zn − y, v∗n − y∗〉 ≥ 0, ∀y ∈ δUX , y∗ ∈ By, n ∈ N ⇒ 〈zn, v∗n〉 − 〈y, v∗n〉+ 〈zn − y,−y∗〉 ≥ 0, ∀y ∈ δUX , y∗ ∈ By, n ∈ N ⇒ 〈zn, v∗n〉 − 〈y, v∗n〉 ≥ 〈zn − y, y∗〉, ∀y ∈ δUX , y∗ ∈ By, n ∈ N ⇒ 〈zn, v∗n〉 − 〈y, v∗n〉 ≥ −(‖zn‖+ δ)M, ∀y ∈ δUX , n ∈ N (by (5.86)) ⇒ 〈zn, v∗n〉 ≥ 〈y, v∗n〉 − (‖zn‖+ δ)M, ∀y ∈ δUX , n ∈ N ⇒ 〈zn, v∗n〉 ≥ δ‖v∗n‖ − (‖zn‖+ δ)M, ∀n ∈ N ⇒ 〈zn, v ∗ n ‖v∗n‖ 〉 ≥ δ − (‖zn‖+δ)M ‖v∗n‖ , ∀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 (v∗γ)γ∈Γ of (v∗n)n∈N, say v∗γ ‖v∗γ‖ 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 n→∞ 〈z − zn, v∗n〉 < −ε, for some ε > 0. Then, for all n sufficiently large, 〈z − zn, v∗n〉 < − ε2 , and so 〈z − zn, v ∗ n ‖v∗n‖〉 < − ε 2‖v∗n‖ . (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∗〉+ 14nα+ 〈z − zn, v∗n〉] ≥ sup {n∈N} [−‖zn‖ · ‖z∗‖+ 14nα+ 〈z − zn, v∗n〉] ≥ lim sup n→∞ [−‖zn‖ · ‖z∗‖+ 14nα+ 〈z − zn, v∗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 [domFA+B ]. Hence PX [domFA+B ] ⊆ domA ∩ domB and thus (5.18) holds. Thus we have PX [domFA+B ] = domA ∩ domB. 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 domA∩ 114 5.2. Maximality of the sum of a (FPV) operator and a full domain operator int domB 6= ∅. Assume that domA ∩ domB ⊆ domB. Then PX [domFA+B ] = domA ∩ domB. 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 domA ∩ int domB 6= ∅. Assume that domA ∩ domB ⊆ domB. Then PX [domFA+B ] = domA ∩ domB. Proof. Combine Fact 5.1.8, Corollary 5.1.18 and Proposition 5.2.1. Alter- natively, 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 domA∩ int domB 6= ∅. Assume that A + NdomB is maximally monotone of type (FPV), and that domA ∩ domB ⊆ domB. Then A + B is maximally monotone. Proof. After translating the graphs if necessary, we can and do assume that 0 ∈ domA ∩ int domB and that (0, 0) ∈ graA ∩ graB. By Fact 5.1.5, domA ⊆ PX(domFA) and domB ⊆ PX(domFB). Hence, ⋃ λ>0 λ ( PX(domFA)− PX(domFB) ) = 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,x∗,y∗} [〈x, z∗〉+ 〈z, x∗〉 − 〈x, x∗〉+ 〈z − x, y∗〉 − ιgraA(x, x∗)− ιgraB(x, y∗)]. (5.44) Assume to the contrary that FA+B(z, z ∗) < 〈z, z∗〉. (5.45) Then (z, z∗) ∈ domFA+B and, by Proposition 5.2.1, z ∈ domA ∩ domB = PX [domFA+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 domFB . By Fact 5.1.8 again and 0 ∈ int domB, 0 ∈ intPX domFB . Then, by [92, Theorem 1.1.2(ii)], we have λz ∈ intPX domFB = int [PX domFB ] . (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 domB. We consider two cases. Case 1 : λz ∈ domA. By (5.44), FA+B(λz, λz ∗) ≥ sup {x∗,y∗} [〈λz, λz∗〉+ 〈λz, x∗〉 − 〈λz, x∗〉+ 〈λz − λz, y∗〉 − ιgraA(λz, x∗)− ιgraB(λz, y∗)] = 〈λz, λz∗〉. Hence (5.47) holds. Case 2 : λz /∈ domA. Using 0 ∈ domA ∩ domB and the convexity of domA ∩ domB (which follows from (5.46)), we obtain λz ∈ domA ∩ domB ⊆ domA ∩ domB. Set Un = λz + 1 n UX , ∀n ∈ N. (5.49) Then Un ∩ dom(A+NdomB) 6= ∅. Since (λz, λz∗) /∈ gra(A+NdomB), λz ∈ Un, and A+NdomB is of type (FPV), there exists (bn, b ∗ n) ∈ gra(A+NdomB) 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 domB and bn → λz, by Fact 5.1.3, there exist N ∈ N and M > 0 such that bn ∈ int domB and sup v∗∈Bbn ‖v∗‖ ≤M, ∀n ≥ N. (5.51) Hence NdomB(bn) = {0} and thus (bn, b∗n) ∈ graA for every n ≥ N . Thus by (5.44), (5.50) and (5.51), FA+B(λz, λz ∗) ≥ sup {v∗∈Bbn} [〈bn, λz∗〉+ 〈λz, b∗n〉 − 〈bn, b∗n〉+ 〈λz − bn, v∗〉] , ∀n ≥ N ≥ sup {v∗∈Bbn} [ λ2〈z, z∗〉+ 〈λz − bn, v∗〉 ] , ∀n ≥ N (by (5.50)) ≥ sup [λ2〈z, z∗〉 −M‖λz − bn‖] , ∀n ≥ N (by (5.51)) ≥ λ2〈z, z∗〉 (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) ∈ graA∩ graB, 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 λ→1− FA+B(λz, λz ∗) ≥ lim λ→1− 〈λz, λz∗〉 = 〈z, z∗〉, (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 domB 6= ∅. Assume that dom ∂f ∩ domB ⊆ domB. Then ∂f + B is maximally monotone. Proof. By Fact 5.1.8 and Fact 5.1.1, ∂f +NdomB = ∂(f + ιdomB). Then by Fact 5.1.12, ∂f +NdomB 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+NdomB = A+NX = A and thus A+NdomB 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, The- orem 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 domA ∩ int domB 6= ∅. Assume that domA ∩ domB ⊆ domB. 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 domA ∩ intC 6= ∅. 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 rela- tion, and let B : X ⇒ X∗ be maximally monotone with domA∩ int domB 6= ∅. Assume that domA∩domB ⊆ domB. Then A+B is maximally mono- tone. 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 domA∩ intC 6= ∅. 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 rela- tion, 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 7→ {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 affir- mative solution when A is a linear relation, B is the subdifferential operator of a proper lower semicontinuous sublinear function, and Rockafellar’s con- straint 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 exam- ple, 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 rela- tion, and let f : X → ]−∞,+∞] be a proper lower semicontinuous convex function with domA∩int dom ∂f 6= ∅. Then A+∂f is maximally monotone. Proof. After translating the graphs if necessary, we can and do assume that 0 ∈ domA ∩ int dom ∂f and that (0, 0) ∈ graA ∩ 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, domA ⊆ PX(domFA) and dom ∂f ⊆ PX(domF∂f ). Hence, ⋃ λ>0 λ ( PX(domFA)− PX(domF∂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,x∗,y∗} [〈x, z∗〉+ 〈z, x∗〉 − 〈x, x∗〉+ 〈z − x, y∗〉 − ιgraA(x, x∗)− ιgra ∂f (x, y∗)]. (5.57) Assume to the contrary that FA+∂f (z, z ∗) + λ < 〈z, z∗〉, (5.58) where λ > 0. Thus by (5.58), (z, z∗) is monotonically related to gra(A+ ∂f). (5.59) We claim that z /∈ domA. (5.60) 123 5.3. Maximality of the sum of a linear relation and a subdifferential operator Indeed, if z ∈ domA, apply (5.59) and Lemma 5.1.20 to get z ∈ dom ∂f . Thus z ∈ domA∩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) ∈ graA ∩ gra ∂f , we have sup [〈0, z∗〉+ 〈z,A0〉 − 〈0, A0〉 + 〈z, ∂f(0)〉] = sup a∗∈A0,b∗∈∂f(0) [〈z, a∗〉+ 〈z, b∗〉] < 〈z, z∗〉. Thus, because A0 is a linear subspace, z ∈ X ∩ (A0)⊥. (5.61) Then, by Proposition 3.2.2(i), we have z ∈ domA. (5.62) Combine (5.60) and (5.62), z ∈ domA\domA. (5.63) Set Un = z + 1 n UX , ∀n ∈ N. (5.64) By (5.63), (z, z∗) /∈ graA and Un ∩ domA 6= ∅. Since z ∈ Un and A is of type (FPV) by Fact 5.1.14, there exists (an, a ∗ n) ∈ graA 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) Then we have an → z. (5.66) Now we claim that z ∈ dom∂f. (5.67) Suppose to the contrary that z 6∈ 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 6∈ 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(βnan) with x∗n 6= z∗ 125 5.3. Maximality of the sum of a linear relation and a subdifferential operator and βnan 6= z such that ‖z − βnan‖ ‖z∗ − x∗n‖ ≥ n (5.70) 〈z − βnan, z∗ − x∗n〉 ‖z − βnan‖ · ‖z∗ − x∗n‖ < −34 . (5.71) By (5.66), ‖z − βnan‖ 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 w∗n ∈ ∂f(βnan) and v∗n ∈ ∂ι[0,an](βnan) such that x∗n = w∗n + v∗n. Then by (5.72), w∗n + v ∗ n → 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), βnan → βz. (5.74) We claim that β ≤ δ < 1. (5.75) 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 v∗n ∈ ∂ι[0,an](βnan), we have 〈v∗n, an − βnan〉 ≤ 0. (5.76) Dividing by (1− βn) on both sides of the above inequality, we have 〈v∗n, an〉 ≤ 0. (5.77) Since (0, 0) ∈ graA, 〈an, a∗n〉 ≥ 0,∀n ∈ N. Then by (5.65), we have 〈z, βna∗n〉+ 〈βnan, z∗〉 − β2n〈an, a∗n〉 ≥ 〈βnz, a∗n〉+ 〈βnan, z∗〉 − βn〈an, a∗n〉 ≥ βn〈z, z∗〉. (5.78) Then by (5.78), 〈z − βnan, βna∗n〉 ≥ 〈βnz − βnan, z∗〉. (5.79) Since graA is a linear subspace and (an, a ∗ n) ∈ graA, (βnan, βna∗n) ∈ graA. By (5.58), we have λ <〈z − βnan, z∗ − w∗n − βna∗n〉 = 〈z − βnan, z∗ − w∗n〉+ 〈z − βnan,−βna∗n〉 127 5.3. Maximality of the sum of a linear relation and a subdifferential operator < −34‖z − βnan‖ · ‖z∗ − w∗n − v∗n‖+ 〈z − βnan, v∗n〉 + 〈z − βnan,−βna∗n〉 (by (5.71)) ≤ −34‖z − βnan‖ · ‖z∗ − w∗n − v∗n‖+ 〈z − βnan, v∗n〉 − 〈βnz − βnan, z∗〉 (by (5.79)). Then λ < 〈z − βnan, v∗n〉 − 〈βnz − βnan, z∗〉. (5.80) Now we consider two cases: Case 1 : (w∗n)n∈N is bounded. By (5.73), (v∗n)n∈N is bounded. By the Banach-Alaoglu Theorem (see [69, Theorem 3.15]), there exist a weak* convergent subnet (v∗γ)γ∈Γ of (v∗n)n∈N, say v∗γ 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∗∞〉 ≥ λ1−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 : (w∗n)n∈N is unbounded. Since (w∗n)n∈N is unbounded and after passing to a subsequence if nec- essary, we assume that ‖w∗n‖ 6= 0,∀n ∈ N and that ‖w∗n‖ → +∞. By the Banach-Alaoglu Theorem again, there exist a weak* convergent subnet (w∗ν)ν∈I of (w∗n)n∈N, say w∗ν ‖w∗ν‖ 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) 6= ∅ and sup y∗∈∂f(y) ‖y∗‖ ≤M, ∀y ∈ ρUX . (5.86) Then by w∗n ∈ ∂f(βnan), we have 〈βnan − y,w∗n − y∗〉 ≥ 0, ∀y ∈ ρUX , y∗ ∈ ∂f(y) 129 5.3. Maximality of the sum of a linear relation and a subdifferential operator ⇒ 〈βnan, w∗n〉 − 〈y,w∗n〉+ 〈βnan − y,−y∗〉 ≥ 0, ∀y ∈ ρUX , y∗ ∈ ∂f(y) ⇒ 〈βnan, w∗n〉 − 〈y,w∗n〉 ≥ 〈βnan − y, y∗〉, ∀y ∈ ρUX , y∗ ∈ ∂f(y) ⇒ 〈βnan, w∗n〉 − 〈y,w∗n〉 ≥ −(‖βnan‖+ ρ)M, ∀y ∈ ρUX (by (5.86)) ⇒ 〈βnan, w∗n〉 ≥ 〈y,w∗n〉 − (‖βnan‖+ ρ)M, ∀y ∈ ρUX ⇒ 〈βnan, w∗n〉 ≥ ρ‖w∗n‖ − (‖βnan‖+ ρ)M ⇒ 〈βnan, w ∗ n ‖w∗n‖〉 ≥ ρ− (‖βnan‖+ ρ)M ‖w∗n‖ , ∀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 6= 0 and thus β > 0. Then by (5.88), 〈z, w∗∞ ≥ ρβ > 0. (5.89) By (5.73) and z ∗ ‖w∗n‖ → 0, we have w∗n ‖w∗n‖ + v∗n ‖w∗n‖ → 0. (5.90) By(5.85), taking the weak∗ limit in (5.90) along the subnet, we obtain v∗ν ‖w∗ν‖ w* ⇁−w∗∞. (5.91) 130 5.3. Maximality of the sum of a linear relation and a subdifferential operator Dividing by ‖w∗n‖ on the both sides of (5.80), we get that λ ‖w∗n‖ < 〈z − βnan, v ∗ n ‖w∗n‖ 〉 − 〈βnz − βnan, z ∗〉 ‖w∗n‖ . (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) By (5.75) and (5.93), 〈z,−w∗∞〉 ≥ 0, (5.94) 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) By Fact 5.1.4, tz ∈ int dom ∂f. (5.97) 131 5.3. Maximality of the sum of a linear relation and a subdifferential operator Set Hn = tz + 1 n UX , ∀n ∈ N. (5.98) Since domA is a linear subspace, tz ∈ domA\domA by (5.63). Then Hn ∩ domA 6= ∅. Since (tz, tz∗) /∈ graA and tz ∈ Hn, A is of type (FPV) by Fact 5.1.14, there exists (bn, b ∗ n) ∈ graA 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)} [〈bn, tz∗〉+ 〈tz, b∗n〉 − 〈bn, b∗n〉+ 〈tz − bn, c∗〉] , ∀n ≥ N ≥ sup {c∗∈∂f(bn)} [ t2〈z, z∗〉+ 〈tz − bn, c∗〉 ] , ∀n ≥ N (by (5.99)) ≥ sup [t2〈z, z∗〉 −K‖tz − bn‖] , ∀n ≥ N (by (5.100)) ≥ t2〈z, z∗〉 (by bn → tz). (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) ∈ graA ∩ 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 t→1− FA+∂f (tz, tz ∗) ≥ lim t→1− 〈tz, tz∗〉 = 〈z, z∗〉, (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 domA ∩ dom ∂f 6= ∅, we have domA = X since domA 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 domA ∩ int dom∂f 6= ∅. Then A+ ∂f is of type (FPV ). Proof. By Theorem 5.3.1, A + ∂f is maximally monotone. Let C be a nonempty closed convex subset ofX, and suppose that dom(A+∂f)∩intC 6= ∅. Let x1 ∈ domA∩ int dom ∂f and x2 ∈ dom(A+ ∂f)∩ intC. 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) + 12δUX ⊆ C. Clearly, x2+λ(x1−x2)+λδUX ⊆ dom f . Thus x2+λ(x1−x2)+ λδ2 UX ⊆ dom f∩C = dom(f + ιC). By Fact 5.1.4, x2 + λ(x1 − x2) + λδ2 UX ⊆ dom∂(f + ιC). Since domA is convex, x2 + λ(x1 − x2) ∈ domA and x2 + λ(x1 − x2) ∈ domA ∩ 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 (FPV ). 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 domA ∩ int dom∂f 6= ∅. Then dom(A+ ∂f) = conv dom(A+ ∂f) = PX domFA+∂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 domA∩ intC 6= ∅. 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 semi- continuous sublinear function, and suppose that domA ∩ int dom ∂f 6= ∅. 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 7→ {x′}, if x ∈ D; ∅, otherwise. Define f : X → ]−∞,+∞] by f(x) = 1 1−‖x‖2 1 , if ‖x‖ < 1; +∞, otherwise. (5.104) 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 ∈ domA ∩ int [dom ∂f ], Theorem 5.3.1 implies that A+ ∂f is max- imally 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 domA = X. Then ∂f + A is maximally monotone.” The domA 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 domA∩domB is closed, domA is convex and Rockafellar’s constraint qualification holds. In Example 5.4.1, domA ∩ dom∂f is not closed by (5.105). Hence we cannot apply for [84, Theo- rem 5.10(η)]. (3) In [86, Corollary 4], Voisei and Zălinescu showed that the sum problem has an affirmative solution when ic(domA) 6= ∅,ic (domB) 6= ∅ and 0 ∈ic [domA− domB]. Since the domA in Example 5.4.1 is a proper 136 5.4. An example and comments dense linear subspace, ic(domA) = ∅. Thus we cannot apply for [86, Corollary 4]. (Given a set C ⊆ X, we define icC by icC = iC, if aff C is closed; ∅, otherwise, where iC [92] is the intrinsic core or relative algebraic interior of C, defined by iC = {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 affir- mative solution when A is a linear relation, B is a maximally monotone operator satisfying Rockafellar’s constraint qualification and domA ∩ domB ⊆ domB. In Example 5.4.1, since domA is a linear subspace, we can take x0 ∈ domA with ‖x0‖ = 1. Thus, by (5.105), we have that x0 ∈ domA ∩ UX = domA ∩ dom ∂f but x0 6∈ UX = dom ∂f. (5.106) Thus domA ∩ 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 Theo- rem 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 domA ∩ intC 6= ∅. 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 domA ∩ int domB 6= ∅. Assume that domA ∩ domB ⊆ domB. 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 ques- tions 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 Fitzpatrick- Phelps 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∗)∈graA 〈a− x∗∗, a∗ − x∗〉 ≥ 0, there exists a bounded net (aα, a ∗ α)α∈Γ in graA 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∗)∈graA (〈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 ∩ ranA 6= ∅, x∗ ∈ U , and (x, x∗) ∈ X ×X∗ is monotonically related to graA ∩ (X × U) it must follow that (x, x∗) ∈ graA. 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 graA = {(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∗) F(z,z∗)(x, x ∗) = F (z + x, z∗ + x∗)− (〈x, z∗〉+ 〈z, x∗〉+ 〈z, z∗〉) = 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 10 = ∞). Assume that x0 ∈ dom f and x∗0 ∈ ∂εf(x0). There exist xε ∈ X,x∗ε ∈ X∗ such that ‖xε − x0‖+ β |〈xε − x0, x∗0〉| ≤ √ ε, x∗ε ∈ ∂f(xε), ‖x∗ε − x∗0‖ ≤ √ ε(1 + β‖x∗0‖), |〈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 max- imally 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∗ [ F(x0,x∗0)(x, x ∗) + 12‖x‖2 + 12‖x∗‖2 ] = 0. 6.2 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 an- swer 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 : domA → X∗ be linear and maximally monotone. As- sume 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∗) 7→ sup (a,a∗)∈graA (〈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, Prob- lem 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 (0, 0) ∈ graA. Let (x∗∗0 , x∗0) ∈ X∗∗ ×X∗. We must show that ΦA(x ∗∗ 0 , x ∗ 0) ≥ 〈x∗∗0 , x∗0〉 (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∗∗ rX. 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 α∈I ‖xα‖ < +∞ (6.3) and 〈xα, x∗0〉 → 〈x∗∗0 , x∗0〉. (6.4) Now we consider two subcases. Subcase 2.1 : There exists α ∈ I, such that (xα, x∗0) ∈ graA. By definition, ΦA(x ∗∗ 0 , x ∗ 0) ≥ 〈xα, x∗0〉+ 〈x∗∗0 , x∗0〉 − 〈xα, x∗0〉 = 〈x∗∗0 , x∗0〉. Hence (6.2) holds. Subcase 2.2 : We have (xα, x ∗ 0) /∈ graA, ∀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) ∈ graA, we have, by the definition of Uε, 0 ∈ ranA ∩ 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 graA 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 , a∗α,ε〉 − 〈aα,ε, a∗α,ε〉 > 〈x∗∗0 − xα, a∗α,ε〉+ 〈xα, x∗0〉. Hence, ΦA(x ∗∗ 0 , x ∗ 0) > 〈x∗∗0 − xα, a∗α,ε〉+ 〈xα, x∗0〉. (6.8) Since a∗α,ε ∈ Uε, there exist tα,ε ∈ [0, 1] and b∗α,ε ∈ UX∗ (6.9) such that a∗α,ε = tα,εx ∗ 0 + εb ∗ α,ε. (6.10) Using (6.8), (6.10), and (6.3), we deduce that ΦA(x ∗∗ 0 , x ∗ 0) > 〈x∗∗0 − xα, tα,εx∗0 + εb∗α,ε〉+ 〈xα, x∗0〉 = tα,ε〈x∗∗0 − xα, x∗0〉+ ε〈x∗∗0 − xα, b∗α,ε〉+ 〈xα, x∗0〉 ≥ tα,ε〈x∗∗0 − xα, x∗0〉 − ε‖x∗∗0 − xα‖+ 〈xα, x∗0〉 145 6.2. Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type ≥ tα,ε〈x∗∗0 − xα, x∗0〉 − ε(‖x∗∗0 ‖+M) + 〈xα, x∗0〉. (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 ∗∗ 0 , x ∗ 0) ≥ −ε(‖x∗∗0 ‖+M) + 〈x∗∗0 , x∗0〉. (6.12) Next, letting ε→ 0 in (6.12), we have ΦA(x ∗∗ 0 , x ∗ 0) ≥ 〈x∗∗0 , x∗0〉. (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 fol- lowing 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 ques- tion posed by Phelps and Simons. Phelps and Simons posed the following question in [63, Section 9, item 2]: Let A : domA → X∗ be linear and maximally monotone. As- sume 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 , a∗0) ∈ graA∗ such that 〈a∗∗0 , a∗0〉 < 0. Then we have sup (a,a∗)∈graA (〈a,−a∗0〉+ 〈a∗∗0 , a∗〉 − 〈a, a∗〉) = sup (a,a∗)∈graA {−〈a, a∗〉} = 0 < 〈−a∗∗0 , a∗0〉, which contradicts that A is of type (NI). Hence A∗ is monotone. “⇐”: Define F : X ×X∗ → ]−∞,+∞] : (x, x∗) 7→ ιgraA(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, v ∗ 0) ∈ X×X∗. Recalling (6.1), note that F(v0,v∗0) : (x, x ∗) 7→ ιgraA(v0 + x, v∗0 + 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,v∗0 )(x, x ∗) + 12‖x‖2 + 12‖x∗‖2 ] = −(F(v0,v∗0 ) + 12‖ · ‖2 + 12‖ · ‖2)∗(0, 0) = −F ∗(v0,v∗0)(y ∗, y∗∗)− 12‖y∗∗‖2 − 12‖y∗‖2. (6.15) 148 6.3. The adjoint of a maximally monotone linear relation Since (x, x∗) 7→ F(v0,v∗0)(x, x∗)+ 12‖x‖2+ 12‖x∗‖2 is coercive, there existM > 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,v∗0 )(an, a ∗ n) + 1 2‖an‖2 + 12‖a∗n‖2 < K + 1 n2 = −F ∗(v0,v∗0 )(y ∗, y∗∗)− 12‖y∗∗‖2 − 12‖y∗‖2 + 1n2 (by (6.15) ) ⇒ F(v0,v∗0)(an, a∗n) + 12‖an‖2 + 12‖a∗n‖2 + F ∗(v0,v∗0)(y ∗, y∗∗) + 12‖y∗∗‖2 + 12‖y∗‖2 < 1n2 (6.17) ⇒ F(v0,v∗0)(an, a∗n) + F ∗(v0,v∗0)(y ∗, y∗∗) + 〈an,−y∗〉+ 〈a∗n,−y∗∗〉 < 1n2 (6.18) ⇒ (y∗, y∗∗) ∈ ∂ 1 n2 F(v0,v∗0 )(an, a ∗ n) (by [92, Theorem 2.4.2(ii)]). (6.19) Set β = 1max{‖y∗‖,‖y∗∗‖}+1 . Then by Fact 6.1.3, there exist sequences (ãn, ã∗n)n∈N in X ×X∗ and (y∗n, y∗∗n )n∈N in X∗ ×X∗∗ such that, ∀n ∈ N, ‖an − ãn‖+ ‖a∗n − ã∗n‖+ β ∣∣∣〈ãn − an, y∗〉+ 〈ã∗n − a∗n, y∗∗〉∣∣∣ ≤ 1n (6.20) max{‖y∗n − y∗‖, ‖y∗∗n − y∗∗‖} ≤ 2n (6.21)∣∣∣〈ãn − an, y∗n〉+ 〈ã∗n − a∗n, y∗∗n 〉∣∣∣ ≤ 1n2 + 1nβ (6.22) (y∗n, y ∗∗ n ) ∈ ∂F(v0,v∗0)(ãn, ã∗n). (6.23) Then, ∀n ∈ N, we have 〈ãn, y∗n〉+ 〈ã∗n, y∗∗n 〉 − 〈an, y∗〉 − 〈a∗n, y∗∗〉 149 6.3. The adjoint of a maximally monotone linear relation = 〈ãn − an, y∗n〉+ 〈an, y∗n − y∗〉+ 〈ã∗n − a∗n, y∗∗n 〉+ 〈a∗n, y∗∗n − y∗∗〉 ≤ ∣∣∣〈ãn − an, y∗n〉+ 〈ã∗n − a∗n, y∗∗n 〉∣∣∣+ |〈an, y∗n − y∗〉|+ |〈a∗n, y∗∗n − y∗∗〉| ≤ 1 n2 + 1 nβ + ‖an‖ · ‖y∗n − y∗‖+ ‖a∗n‖ · ‖y∗∗n − y∗∗‖ (by (6.22)) ≤ 1 n2 + 1 nβ + (‖an‖+ ‖a∗n‖) ·max{‖y∗n − y∗‖, ‖y∗∗n − y∗∗‖} ≤ 1 n2 + 1 nβ + 2 n M (by (6.16) and (6.21)). (6.24) By (6.20), ∀n ∈ N, we have ∣∣‖an‖ − ‖ãn‖∣∣+ ∣∣‖a∗n‖ − ‖ã∗n‖∣∣ ≤ 1n . (6.25) Thus by (6.16), ∀n ∈ N, we have ∣∣‖an‖2 − ‖ãn‖2∣∣+ ∣∣∣‖a∗n‖2 − ‖ã∗n‖2∣∣∣ = ∣∣‖an‖ − ‖ãn‖∣∣(‖an‖+ ‖ãn‖) + ∣∣∣‖a∗n‖ − ‖ã∗n‖∣∣∣ (‖a∗n‖+ ‖ã∗n‖) ≤ 1 n ( 2‖an‖+ 1n ) + 1 n ( 2‖a∗n‖+ 1n ) (by (6.25)) ≤ 1 n (2M + 2 n ) = 2 n M + 2 n2 . (6.26) Similarly, by (6.21), for all n ∈ N, we have ∣∣‖y∗n‖2 − ‖y∗‖2∣∣ ≤ 4n‖y∗‖+ 4n2 ≤ 4nβ + 4n2 ,∣∣‖y∗∗n ‖2 − ‖y∗∗‖2∣∣ ≤ 4n‖y∗∗‖+ 4n2 ≤ 4nβ + 4n2 . (6.27) Thus, ∀n ∈ N, F(v0,v∗0 )(ãn, ã ∗ n) + F ∗ (v0,v∗0 ) (y∗n, y ∗∗ n ) + 1 2‖ãn‖2 + 12‖ã∗n‖2 + 12‖y∗n‖2 + 12‖y∗∗n ‖2 150 6.3. The adjoint of a maximally monotone linear relation = [ F(v0,v∗0 )(ãn, ã ∗ n) + F ∗ (v0,v∗0 ) (y∗n, y ∗∗ n ) + 1 2‖ãn‖2 + 12‖ã∗n‖2 + 12‖y∗n‖2 + 12‖y∗∗n ‖2 ] − [F(v0,v∗0)(an, a∗n) + 12‖an‖2 + 12‖a∗n‖2 + F ∗(v0,v∗0 )(y∗, y∗∗) + 12‖y∗∗‖2 + 12‖y∗‖2 ] + [ F(v0,v∗0)(an, a ∗ n) + 1 2‖an‖2 + 12‖a∗n‖2 + F ∗(v0,v∗0 )(y ∗, y∗∗) + 12‖y∗∗‖2 + 12‖y∗‖2 ] < [ F(v0,v∗0 )(ãn, ã ∗ n) + F ∗ (v0,v∗0 ) (y∗n, y ∗∗ n )− F(v0,v∗0 )(an, a∗n)− F ∗(v0,v∗0 )(y ∗, y∗∗) ] + 12 [ ‖ãn‖2 + ‖ã∗n‖2 − ‖an‖2 − ‖a∗n‖2 ] + 12 [‖y∗n‖2 + ‖y∗∗n ‖2 − ‖y∗∗‖2 − ‖y∗‖2]+ 1n2 (by (6.17)) ≤ [ 〈ãn, y∗n〉+ 〈ã∗n, y∗∗n 〉 − 〈an, y∗〉 − 〈a∗n, y∗∗〉 ] (by (6.23)) + 12 (∣∣‖ãn‖2 − ‖an‖2∣∣+ ∣∣∣‖ã∗n‖2 − ‖a∗n‖2∣∣∣) + 12 (∣∣‖y∗n‖2 − ‖y∗‖2∣∣+ ∣∣‖y∗∗n ‖2 − ‖y∗∗‖2∣∣)+ 1n2 ≤ 1 n2 + 1 nβ + 2 n M + 1 n M + 1 n2 + 4 nβ + 4 n2 + 1 n2 (by (6.24), (6.26) and (6.27)) = 7 n2 + 5 nβ + 3 n M. (6.28) By (6.23), (6.14), and [92, Theorem 3.2.4(vi)&(ii)], there exists a sequence (z∗n, z∗∗n )n∈N in (graA)⊥ and such that (y∗n, y ∗∗ n ) = (ã ∗ n, ãn) + (z ∗ n, z ∗∗ n ), ∀n ∈ N. (6.29) 151 6.3. The adjoint of a maximally monotone linear relation Since A∗ is monotone and (z∗∗n , z∗n) ∈ gra(−A∗), it follows from (6.29) that, ∀n ∈ N, 〈y∗n, y∗∗n 〉 − 〈y∗n, ãn〉 − 〈y∗∗n , ã∗n〉+ 〈ã∗n, ãn〉 = 〈y∗n − ã∗n, y∗∗n − ãn〉 (6.30) = 〈z∗n, z∗∗n 〉 ≤ 0 ⇒ 〈y∗n, y∗∗n 〉 ≤ 〈y∗n, ãn〉+ 〈y∗∗n , ã∗n〉 − 〈ã∗n, ãn〉. Then by (6.14) and (6.23), we have 〈ã∗n, ãn〉 = F(v0,v∗0)(ãn, ã∗n) and, ∀n ∈ N, 〈y∗n, y∗∗n 〉 ≤ 〈y∗n, ãn〉+ 〈y∗∗n , ã∗n〉 − F(v0,v∗0)(ãn, ã∗n) = F ∗(v0,v∗0)(y ∗ n, y ∗∗ n ). (6.31) By (6.28) and (6.31), ∀n ∈ N, we have F(v0,v∗0)(ãn, ã ∗ n) + 〈y∗n, y∗∗n 〉+ 12‖ãn‖2 + 12‖ã∗n‖2 + 12‖y∗n‖2 + 12‖y∗∗n ‖2 < 7 n2 + 5 nβ + 3 n M ⇒ F(v0,v∗0 )(ãn, ã∗n) + 12‖ãn‖2 + 12‖ã∗n‖2 < 7n2 + 5nβ + 3nM. (6.32) Thus by (6.32), inf (x,x∗)∈X×X∗ [ F(v0,v∗0 )(x, x ∗) + 12‖x‖2 + 12‖x∗‖2 ] ≤ 0. (6.33) By (6.14), inf (x,x∗)∈X×X∗ [ F(v0,v∗0 )(x, x ∗) + 12‖x‖2 + 12‖x∗‖2 ] ≥ 0. (6.34) 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,v∗0 )(x, x ∗) + 12‖x‖2 + 12‖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 Theo- rem 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 re- sult. 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 7→ {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, domA∗ = {z ∈ X∗∗ | z is absolutely continuous, z(1) = 0, z′ ∈ X∗} ⊆ X A∗z = −z′,∀z ∈ domA∗, 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 ques- tion: 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 F12F2, is the function defined on X ×X∗ by F12F2 : (x, x ∗) 7→ inf y∗∈X∗ F1(x, x ∗ − y∗) + F2(x, y∗). In this chapter, we study the properties of FA2FB for two maximally mono- tone operators A and B. We also consider the connection between FA2FB 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 sup- pose that [domA− domB] 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 ∗) = 2q∗A+( 1 2x ∗+12A ∗x) = 12q ∗ A+ (x∗+A∗x), ∀(x, x∗) ∈ X×X. (7.1) If ranA+ is closed, then dom q ∗ A+ = ranA+. Proof. By Proposition 3.1.3(ix), domA∗∩X = X. Hence for every (x, x∗) ∈ X ×X∗, FA(x, x ∗) = sup y∈X [〈x,Ay〉+ 〈y, x∗〉 − 〈y,Ay〉] = 2 sup y∈X [〈y, 12x∗ + 12A∗x〉 − qA+(y)] = 2q∗A+( 1 2x ∗ + 12A ∗x) = 12q ∗ A+ (x∗ +A∗x). (7.2) By [92, Proposition 2.4.4(iv) and Theorem 2.3.3], ran ∂qA+ ⊆ dom∂q∗A+ . (7.3) By Proposition 3.2.10, ran ∂qA+ = ranA+. Then by (7.3), ranA+ ⊆ dom ∂q∗A+ ⊆ dom q∗A+ (7.4) 157 7.1. Auxiliary results Then by the Brøndsted-Rockafellar Theorem (see [92, Theorem 3.1.2]), ranA+ ⊆ dom∂q∗A+ ⊆ dom q∗A+ ⊆ ranA+. By the assumption that ranA+ is closed, we have ranA+ = dom ∂q ∗ A+ = dom q∗A+ . 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∗)∈graNC [〈x, c∗〉+ 〈c, x∗〉 − 〈c, c∗〉] = sup (c,c∗)∈graNC ,k≥0 [〈x, kc∗〉+ 〈c, x∗〉 − 〈c, kc∗〉] = sup (c,c∗)∈graNC ,k≥0 [k(〈x, c∗〉 − 〈c, c∗〉) + 〈c, x∗〉] (7.5) By (7.5), (x, x∗) ∈ domFNC ⇒ sup (c,c∗)∈graNC [〈x, c∗〉 − 〈c, c∗〉] ≤ 0 ⇔ inf (c,c∗)∈graNC [−〈x, c∗〉+ 〈c, c∗〉] ≥ 0 ⇔ inf (c,c∗)∈graNC [〈c− x, c∗ − 0〉] ≥ 0 ⇔ (x, 0) ∈ graNC (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 ᵀ : X∗ ×X : (x∗, x) 7→ 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 F ∗ᵀA = 〈·, ·〉 on graA and { x ∈ X | ∃x∗ ∈ X∗ such that F ∗A(x∗, x) = 〈x, x∗〉 } ⊆ conv(domA). Fact 7.1.4 (See [92, Theorem 2.4.14].) Let f : X → ]−∞,+∞] be a sub- linear 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 Zălinescu) (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 , (F12F2)(x, y) > −∞ 159 7.1. Auxiliary results and that ⋃ λ>0 λ [PX domF1 − PX domF2] is a closed subspace of X. Then for every (x∗, y∗) ∈ X ×X∗, (F12F2) ∗(x∗, y∗) = min w∗∈X∗ [F ∗1 (x ∗ − w∗, y∗) + F ∗2 (w∗, y∗)] . 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 graA and graB are closed. Assume that domA− domB is closed. Then (A+B)∗ = A∗ +B∗. Proof. We have ιgra(A+B) = ιgraA2ιgraB. (7.9) Let (x∗∗, x∗) ∈ X∗∗ × X∗. Since graA and graB are closed convex, ιgraA and ιgraB 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∗graA(y ∗, x∗∗) + ι∗graB(−x∗ − y∗, x∗∗) = ι(graA)⊥(y ∗, x∗∗) + ι(graB)⊥(−x∗ − y∗, x∗∗) = ιgraA∗(x ∗∗,−y∗) + ιgraB∗(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 [domA− domB] is a closed subspace of X. Set E = { x ∈ X | ∃x∗ ∈ X∗ such that F ∗A(x∗, x) = 〈x, x∗〉 } and F = { x ∈ X | ∃x∗ ∈ X∗ such that F ∗B(x∗, x) = 〈x, x∗〉 } . Then ⋃ λ>0 λ [domA− domB] = ⋃ λ>0 λ [E − F ] . Moreover, if A and B are of type (FPV), then we have ⋃ λ>0 λ [domA− domB] = ⋃ λ>0 λ [PX domFA − PX domFB ] . Proof. Using Fact 7.1.3, we see that ⋃ λ>0 λ [domA− domB] ⊆ ⋃ λ>0 λ [E − F ] ⊆ ⋃ λ>0 λ [ conv(domA)− conv(domB) ] ⊆ ⋃ λ>0 λ [ conv(domA)− conv(domB) ] 161 7.1. Auxiliary results = ⋃ λ>0 λ[conv(domA− domB)] ⊆ ⋃ λ>0 λ [conv(domA− domB)] = ⋃ λ>0 λ [domA− domB] (using the assumption). Hence ⋃ λ>0 λ [domA− domB] = ⋃ 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 [domA− domB] ⊆ ⋃ λ>0 λ [PX domFA − PX domFB ] ⊆ ⋃ λ>0 λ [ domA− domB] ⊆ ⋃ λ>0 λ [ domA− domB] ⊆ ⋃ λ>0 λ [domA− domB] = ⋃ λ>0 λ [domA− domB] (using the assumption). Corollary 7.1.8 Let A,B : X ⇒ X∗ be maximally monotone linear rela- tions, and suppose that [domA− domB] is a closed subspace. Then ⋃ λ>0 λ [PX domFA − PX domFB ] = [domA− domB] = ⋃ λ>0 λ [ PX domF ∗ᵀ A − PX domF ∗ᵀB ] . 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 [domA− C] is a closed subspace. Then ⋃ λ>0 λ [PX domFA − PX domFNC ] = ⋃ λ>0 λ [domA− C] = ⋃ λ>0 λ [ PX domF ∗ᵀ A − PX domF ∗ᵀNC ] . 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 domA∩domB 6= ∅. Then FA+B ≤ FA2FB. Proof. Let (x, x∗) ∈ X ×X∗ and y∗ ∈ X∗. Then we have FA(x, y ∗) + FB(x, x∗ − y∗) = sup (a,a∗)∈graA [〈a, y∗〉+ 〈x, a∗〉 − 〈a, a∗〉] + sup (b,b∗)∈graB [〈b, x∗ − y∗〉+ 〈x, b∗〉 − 〈b, b∗〉] = sup (a,a∗)∈graA,(b,b∗)∈graB [〈a, y∗〉+ 〈x, a∗〉 − 〈a, a∗〉+ 〈b, x∗ − y∗〉+ 〈x, b∗〉 − 〈b, b∗〉] ≥ sup (a,a∗)∈graA,(a,b∗)∈graB [〈a, y∗〉+ 〈x, a∗〉 − 〈a, a∗〉+ 〈a, x∗ − y∗〉+ 〈x, b∗〉 − 〈a, b∗〉] = sup (a,a∗)∈graA,(a,b∗)∈graB [〈a, x∗〉+ 〈x, a∗ + b∗〉 − 〈a, a∗ + b∗〉] = FA+B(x, x ∗). (7.11) 163 7.1. Auxiliary results Then infy∗∈X∗ [FA(x, y∗) + FB(x, x∗ − y∗)] ≥ FA+B(x, x∗) and thus FA2FB(x, x ∗) ≥ FA+B(x, x∗). We now discover more properties of FA2FB . 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 sup- pose that ⋃ λ>0 λ [domA− domB] is a closed subspace of X. Then FA2FB is proper, norm×weak∗ lower semicontinuous and convex, and the partial in- fimal convolution is exact everywhere. Proof. Define F1, F2 : X ×X∗ → ]−∞,+∞] by F1 : (x, x ∗) 7→ F ∗A(x∗, x), F2 : (x, x∗) 7→ F ∗B(x∗, x). Since FA, FB is norm-weak ∗ lower semicontinuous, F ∗1 (x ∗, x) = FA(x, x∗), F ∗2 (x ∗, x) = FB(x, x∗), ∀(x, x∗) ∈ X ×X∗. (7.12) Take (x, x∗) ∈ X ×X∗. By Fact 5.1.5, ( F12F2 ) (x, x∗) ≥ 〈x, x∗〉 > −∞. 164 7.1. Auxiliary results In view of Lemma 7.1.7, ⋃ λ>0 λ [PX domF1 − PX domF2] = ⋃ λ>0 λ [domA− domB] is a closed subspace. By Fact 7.1.5 and (7.12), ( F12F2 )∗ (x∗, x) = min y∗∈X∗ [F ∗1 (x ∗ − y∗, x) + F ∗2 (y∗, x)] = min y∗∈X∗ [FA(x, x ∗ − y∗) + FB(x, y∗)] = ( FA2FB ) (x, x∗). Hence FA2FB 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 F ∗A : (x ∗, x) 7→ ιgraA(x, x∗) + 〈x, x∗〉. Proof. Define g : X × X∗ → ]−∞,+∞] : (x, x∗) 7→ 〈x, x∗〉 + ιgraA(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 F ∗ᵀA = 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 sup- pose that ⋃ λ>0 λ [PX domFA − PX domFB ] is a closed subspace of X. As- sume that FA2FB = 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 FA2FB(x, x ∗) = inf y∗∈X∗ [FA(x, y ∗) + FB(x, x∗ − y∗)] ≥ 〈x, x∗〉. (7.14) By (7.14) and the assumption that FA2FB = 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 domA∩domB 6= ∅. By Fact 7.1.10 FA2FB ≥ FA+B . It naturally raises a question: Does the equality always hold under the Rockafellar’s constraint qualification: domA ∩ int domB 6= ∅ (which was also asked by the referee of [90])? The equality has a far-reaching meaning. If this were true, then Propo- sition 7.1.13 would directly solve the sum problem in the affirmative. How- ever, in general, it cannot hold. The easiest example probably is [10, Ex- ample 4.7] by Bauschke, McLaren and Sendov on two projection operators in one dimensional space. Now we give another counterexample on a max- imally monotone linear relation and the subdifferential of a proper lower semicontinuous sublinear function, which thus implies that we cannot ap- proach the maximality of the sum of a linear relation A and the subdif- 166 7.1. Auxiliary results ferential of a proper lower semicontinuous sublinear function f by showing FA2F∂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∂f2FId(x, x ∗) = ‖x‖+ 0, if ‖x+ x∗‖ ≤ 1; 1 4‖x+ x∗‖2 − 12‖x+ x∗‖+ 14 , if ‖x+ x∗‖ > 1. (7.15) We also have F∂f+Id 6= F∂f2FId when X = R. Proof. By [10, Example 3.10 and Example 3.3], we have FId(x, x ∗) = 14‖x+ x∗‖2 (7.16) F∂f (x, x ∗) = ‖x‖ + ιBX (x∗), ∀(x, x∗) ∈ X ×X. (7.17) Note that ∂f(x) = BX , if x = 0; { x‖x‖}, otherwise. (7.18) NBX (x) = 0, if ‖x‖ < 1; [0,∞[ · x, if ‖x‖ = 1; ∅, otherwise. (7.19) 167 7.1. Auxiliary results Indeed, clearly ∂f(0) = BX . Assume x 6= 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∂f2FId)(x, x ∗) = inf y∗ [‖x‖+ ιBX (y∗) + 14‖x+ x∗ − y∗‖2] = ‖x‖+ 14‖x+ x∗‖2 + 12 infy∗ [〈x+ x∗, y∗〉+ ιBX (y∗) + 12‖y∗‖2] . (7.20) We consider two cases: Case 1 : ‖x+ x∗‖ ≤ 1. Then we directly obtain that inf y∗ [〈x+ x∗, y∗〉+ ιBX (y∗) + 12‖y∗‖2] = −12‖x+ x∗‖2. And thus, F∂f2FId(x, x ∗) = ‖x‖. Case 2 : ‖x+x∗‖ > 1. SinceK : y∗ ∈ X → 〈x+x∗, y∗〉+ιBX (y∗)+ 12‖y∗‖2 is convex, y∗0 is a minimizer of K if and only if 0 ∈ x+ x∗ + y∗0 +NBX (y∗0). Since ‖x + x∗‖ > 1, by (7.19), ‖y∗0‖ = 1. Thus by (7.19) again, there exists ρ > 0 such that 0 = x + x∗ + y∗0 + ρy ∗ 0. Then we have ρ + 1 = ‖x + x∗‖ and y∗0 = − x+x ∗ ‖x+x∗‖ . Thus infK = K(y ∗ 0) = −‖x + x∗‖ + 12 . Then F∂f2FId(x, x ∗) = ‖x‖+ 14‖x+ x∗‖2 − 12‖x+ x∗‖+ 14 . Hence (7.15) holds. 168 7.1. Auxiliary results In order to show F∂f+Id 6= F∂f2FId, we consider the case when X = R. Now we consider the point (−1, 4). Then by (7.15), (F∂f2FId)(−1, 4) = 1 + 1 = 2. (7.21) On the other hand, F∂f+Id(−1, 4) = sup x∈R [〈x, 4〉 + 〈−1, x+ ∂f(x)〉 − 〈x, ∂f(x) + x〉] = sup x∈R [〈x, 3〉 + 〈−1, ∂f(x)〉 − 〈x, ∂f(x) + x〉] = sup x∈R [〈x, 3〉+ 〈−1, ∂f(x)〉 − |x| − |x|2] (by Fact 7.1.4(i)) = max { sup x>0 [〈x, 3〉 + 〈−1, ∂f(x)〉 − |x| − |x|2] , sup x=0 [〈x, 3〉+ 〈−1, ∂f(x)〉 − |x| − |x|2] , sup x<0 [〈x, 3〉+ 〈−1, ∂f(x)〉 − |x| − |x|2] } = max { sup x>0 [〈x, 3〉 − 1− |x| − |x|2] , 1, sup x<0 [〈x, 3〉 + 1− |x| − |x|2] } (by (7.18)) = max { sup x>0 [〈x, 3〉 − 1− x− |x|2] , 1, 1} = max { sup x>0 [ 2x− 1− |x|2] , 1} = max{0, 1} = 1 6= 2 = F∂f2FId(−1, 4) (by (7.21)). Hence F∂f+Id 6= F∂f2FId. 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 [domA− domB] is closed. Then FA+B = FA2FB. Proof. Let (z, z∗) ∈ X ×X∗. By Fact 7.1.10, it suffices to show that FA+B(z, z ∗) ≥ (FA2FB)(z, z∗). (7.22) If (z, z∗) /∈ domFA+B , then (7.22) clearly holds. Now assume that (z, z∗) ∈ domFA+B . Then FA+B(z, z ∗) = sup {x,x∗,y∗} [〈x, z∗〉+ 〈z, x∗〉 − 〈x, x∗〉+ 〈z − x, y∗〉 − ιgraA(x, x∗) − ιgraB(x, y∗)]. (7.23) Let Y = X∗ and define F,K : X ×X∗ × Y → ]−∞,+∞] respectively by F :(x, x∗, y∗) ∈ X ×X∗ × Y 7→ 〈x, x∗〉+ ιgraA(x, x∗) 170 7.2. Fitzpatrick function of the sum of two linear relations K :(x, x∗, y∗) ∈ X ×X∗ × Y 7→ 〈x, y∗〉+ ιgraB(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 semicontin- uous and convex, and domF − domK = [domA− domB]×X∗ × Y is a closed subspace. Thus by Fact 3.1.2 and (7.24), there exist (z∗0 , z ∗∗ 0 , z ∗∗ 1 ) ∈ X∗ × X∗∗ × Y ∗ such that FA+B(z, z ∗) = F ∗(z∗ − z∗0 , z − z∗∗0 , z − z∗∗1 ) +K∗(z∗0 , z∗∗0 , z∗∗1 ) = F ∗(z∗ − z∗0 , z, 0) +K∗(z∗0 , 0, z) (by (z, z∗) ∈ domFA+B) = FA(z, z ∗ − z∗0) + FB(z, z∗0) ≥ (FA2FB)(z, z∗). Thus (7.22) holds and hence FA+B = FA2FB . 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 rela- 171 7.2. Fitzpatrick function of the sum of two linear relations tions, and suppose that [domA− domB] 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 The- orem 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., S : domS → `2(N) : y 7→ ( 1 2yn + ∑ i<n yi ) n∈N , (7.25) with domS = { y = (yn) ∈ `2(N) ∣∣∣∣ ∑ i≥1 yi = 0, (∑ i≤n yi ) n∈N ∈ `2(N) } . We explicitly compute the Fitzpatrick functions FS+S∗ , FS , FS∗ , and show that FS+S∗ 6= FS2FS∗ even though S, S∗ are linear maximally mono- tone with domS − domS∗ being a dense linear subspace in `2(N). Lemma 7.2.3 Let X be a reflexive space and S : domS → X∗ be a maxi- mally monotone skew linear operator. Then FS = ιgra(−S∗) and F ∗ᵀS∗ = FS∗ = ιgraS∗ + 〈·, ·〉. 172 7.2. Fitzpatrick function of the sum of two linear relations Proof. By Proposition 7.1.12, F ∗S = (ιgraS) ᵀ. Then FS = ( F ∗ᵀS )∗ᵀ = ( ιgraS )∗ᵀ = ( ιᵀgraS )∗ = ( ιgraS−1 )∗ = ι(graS−1)⊥ = ιgra(−S∗). (7.26) From Fact 3.2.12, gra(−S) ⊆ graS∗, we have FS∗ ≥ F(−S) = ιgra−(−S)∗ = ιgraS∗ , this shows that domFS∗ ⊆ graS∗. By the Brezis-Browder theorem (Fact 3.2.13) and Fact 5.1.5, FS∗(x, x ∗) = 〈x, x∗〉 ∀(x, x∗) ∈ graS∗. Hence FS∗ = ιgraS∗ + 〈·, ·〉. Again by Proposition 7.1.12, F ∗ᵀS∗ = ιgraS∗ + 〈·, ·〉. 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∗) FS2FS∗(x, x ∗) = 1 2s 2, if (x, x∗) ∈ domS∗ × {0}with s =∑i≥1 xi; ∞ otherwise. (7.27) Consequently, FS2FS∗ 6= 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 (FS2FS∗)(x, x ∗) =∞, x∗ 6= 0. (7.30) It follows from Lemma 7.2.3 that (FS2FS∗)(x, 0) = inf y∗∈H {FS(x, y∗) + FS∗(x,−y∗)} = inf y∗∈H {ιgra(−S∗)(x, y∗) + ιgraS∗(x,−y∗) + 〈x,−y∗〉} = inf y∗∈H {ιgraS∗(x,−y∗) + 〈x,−y∗〉}. (7.31) Thus, FS2FS∗(x, 0) = ∞ if x /∈ domS∗. Now suppose x ∈ domS∗ and s = ∑ i≥1 xi. Then by (7.31) and Proposition 3.3.6, we have FS2FS∗(x, 0) = 〈x, S∗x〉 = 12s2. Combine the results above, (7.27) holds. Since domS∗ 6= H, FS2FS∗ 6= 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 (domA = domA∗ and FA = F ∗ᵀ A ) . Let A = S∗ with S defined as in Example 3.3.1. Lemma 7.2.3 shows that FA = F ∗ᵀ A , but A ∗ = S 6= −S∗ = −A by Proposition 3.3.5. Hence the requirement domA = domA∗ 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 FT2FT ∗ 6= 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 opera- tors, 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 q∗V+(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 q∗V+(z) =∞, if z /∈ span{e}. Now suppose that z = le for some l ∈ R. By Example 3.4.4(iv), q∗V+(z) = sup x∈H {〈x, z〉 − qV+(x)} = sup x∈H {〈x, le〉 − 14〈x, e〉2} = l2 = 〈le, e〉2 = 〈z, e〉2. 175 7.2. Fitzpatrick function of the sum of two linear relations Hence q∗V+(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∗) + 12〈x+ V ∗y∗, e〉2, FT ∗(x, y ∗) = FV ∗(y∗, x) = ιspan{e}(x+ V y∗) + 12 〈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) = 12q ∗ V ∗+ (x+ V ∗∗y∗) = 12q ∗ V+ (x+ V y∗) = ιspan{e}(x+ V y∗) + 12〈x+ V y∗, e〉2. Remark 7.2.7 Theorem 7.2.8 below gives another example showing that FT+T ∗ 6= FT2FT ∗ while T, T ∗ are maximally monotone, and domT − domT ∗ is a dense subspace in L2[0, 1]. This again shows that the assump- tion that domA − domB 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+T ∗(x, x ∗) = ιH×{0}(x, x∗), ∀(x, x∗) ∈ H ×H (FT2FT ∗)(x, x ∗) = 1 2 [ x(1)2 + x(0)2 ] , if (x, x∗) ∈ C × {0}; ∞, otherwise. (7.33) Consequently, FT2FT ∗ 6= FT+T ∗ . Proof. By Theorem 3.4.5(i) and Example 3.4.4(iii), (T + T ∗)y = 0, ∀y ∈ domT ∩ domT ∗ = {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∈domT∩domT ∗ 〈x∗, y〉 = sup y∈H 〈x∗, y〉 = ι{0}(x∗) = ιH×{0}(x, x∗). (7.35) By Fact 7.1.10, we have ( FT2FT ∗ ) (x, x∗) =∞, ∀ x∗ 6= 0. (7.36) When x∗ = 0, by Lemma 7.2.6, ( FT2FT ∗ ) (x, 0) = inf y∗∈H {FT (x, y∗) + FT ∗(x,−y∗)} (7.37) = inf y∗∈H {ιspan{e}(x+ V ∗y∗) + 12〈x+ V ∗y∗, e〉2 + ιspan{e}(x− V y∗) + 12 〈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, (FT2FT ∗)(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 ( FT2FT ∗ ) (x, 0) = 12〈x+ V ∗x′, e〉2 + 12〈x− V x′, e〉2 = 12x(1) 2 + 12x(0) 2 = 12 [ x(1)2 + x(0)2 ] . Thus, (7.33) holds. Consequently, FT2FT ∗ 6= FT+T ∗ . 7.3 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 domA ∩ intC 6= ∅. Then FA+NC = FA2FNC . 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 ∗) ≥ (FA2FNC )(z, z∗). (7.38) By Corollary 5.3.4, PX [domFA+NC ] ⊆ [dom(A+NC)] ⊆ C. Thus, (7.38) holds if z /∈ C. Now assume that z ∈ C. Set g : X ×X∗ → ]−∞,+∞] : (x, x∗) 7→ 〈x, x∗〉+ ιgraA(x, x∗). (7.39) By Fact 3.2.8, g is convex. Hence, h = g + ιC×X∗ (7.40) is convex as well. Let c0 ∈ domA ∩ intC, (7.41) and let c∗0 ∈ Ac0. Then (c0, c∗0) ∈ graA∩(intC×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∗〉 − ιgraA(x, x∗) − ιgraNC (x, c∗) ] ≥ sup (x,x∗) [〈x, z∗〉+ 〈z, x∗〉 − 〈x, x∗〉 − ιgraA(x, x∗)− ιC×X∗(x, x∗)] 179 7.4. Discussion = sup (x,x∗) [〈x, z∗〉+ 〈z, x∗〉 − h(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 6= y∗∗. Clearly, FA+NC (z, z∗) = +∞≥ (FA2FNC )(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) ≥ FA2(ιC + ι∗C) = (FA2FNC )(z, z ∗) (by Fact 7.1.2). Hence (7.38) holds and thus FA+NC = FA2FNC . 7.4 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[ domA−⋃λ>0 λC] is a closed subspace of X. Is it necessarily true that FA+NC = FA2FNC? 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 F11F2 is defined on X × Y by F11F2 : (x, y) 7→ inf u∈X {F1(u, y) + F2(x− u, y)}. 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 λ [PX∗ domF1 − PX∗ domF2] is a closed subspace of X∗. Is F11F2 necessarily a BC–function? 8.1 Auxiliary results Fact 8.1.1 (Banach and Mazur) (See [44, Theorem 5.17]).) Every sep- arable 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 graA. 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 ranL. 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 ∈ domA. Fact 8.1.3 (Simons and Zălinescu) (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 , (F11F2)(x, y) > −∞ and that ⋃ λ>0 λ [PY domF1 − PY domF2] is a closed subspace of Y . Then for every (x∗, y∗) ∈ X∗ × Y ∗, (F11F2) ∗(x∗, y∗) = min u∗∈Y ∗ [F ∗1 (x ∗, u∗) + F ∗2 (x ∗, y∗ − u∗)] . 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 ranL closed and ranL∗ = Y ∗. Let A : X ⇒ X∗ be monotone with domA ⊆ ranL such that graA 6= ∅. Then A is maximally monotone if and only if A is ranL– 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 ranL∗ = 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∗)−1TL−1 is maximally monotone. Proof. Let A = (L∗)−1TL−1. Then domA ⊆ ranL. Since L is an iso- morphism into X, ranL is closed. By Fact 8.1.5, ranL∗ = Y ∗. Hence gra(L∗)−1TL−1 6= ∅ if and only if graT 6= ∅. Clearly, A is monotone. Since (0, (ranL)⊥) ⊆ gra(L∗)−1, A = (L∗)−1TL−1 is ranL–saturated. By 183 8.1. Auxiliary results Fact 8.1.4, A = (L∗)−1TL−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∗)−1TL−1 is maximally monotone but is not of type (D). Proof. By Corollary 8.1.6, (L∗)−1TL−1 is maximally monotone. By Fact 6.1.5 or Corollary 6.2.2 , there exists (y∗∗0 , y ∗ 0) ∈ Y ∗∗ × Y ∗ such that sup (b,b∗)∈gra T {〈y∗∗0 , b∗〉+ 〈y∗0 , b〉 − 〈b, b∗〉} < 〈y∗∗0 , y∗0〉. (8.2) By Fact 8.1.5, there exists x∗0 ∈ X∗ such that L∗x∗0 = y∗0 . Let A = (L∗)−1TL−1. Then we have sup (a,a∗)∈graA {〈L∗∗y∗∗0 , a∗〉+ 〈x∗0, a〉 − 〈a, a∗〉} = sup (Ly,a∗)∈graA {〈y∗∗0 , L∗a∗〉+ 〈x∗0, Ly〉 − 〈Ly, a∗〉} = sup (Ly,a∗)∈graA {〈y∗∗0 , L∗a∗〉+ 〈L∗x∗0, y〉 − 〈y, L∗a∗〉} = sup (Ly,a∗)∈graA {〈y∗∗0 , L∗a∗〉+ 〈y∗0 , y〉 − 〈y, L∗a∗〉} = sup (y,y∗)∈graT {〈y∗∗0 , y∗〉+ 〈y∗0 , y〉 − 〈y, y∗〉} (by (Ly, a∗) ∈ graA⇔ (y, L∗a∗) ∈ graT ) 184 8.2. Main construction < 〈y∗∗0 , y∗0〉 (by (8.2)) = 〈L∗∗y∗∗0 , x∗0〉. (8.3) Thus A is not type (NI) and hence A = (L∗)−1TL−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∗)∈graA {〈(x∗0, x0), (x, x∗)〉 − 〈x, x∗〉} = sup (x,x∗)∈graA 〈(x∗0, x0), (x, x∗)〉 = ι(graA)⊥(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 ranA ⊆ 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 graT = {(−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 func- tion. Set F = f ⊕ f∗ on X ×X∗. Then the following hold. (i) T is maximally monotone. (ii) graT ∗ = {(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 domT ∩ int dom ∂f 6= ∅, 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∗(domFT )− PX∗(domF ) ) = X∗. Assume that there exists (v0, v ∗ 0) ∈ X ×X∗ such that f∗(v∗0) + f ∗∗(v0 −A∗v∗0) < 〈v0, v∗0〉. (8.7) Then FT1F is not a BC–function. (viii) Assume that [ ranA−⋃λ>0 λdom f] is a closed subspace of X and that ∅ 6= dom f∗∗ ◦ A∗|X∗ " {e}⊥. Then T + ∂f is not of type (D). Proof. (i): Now we claim that Px∗ = 〈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∗ = Px∗ + Sx∗ = Ax∗ ∈ ranA ⊆ 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 graT . 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) By Fact 3.2.9 again, κ〈z∗, e〉 = 〈−Sz∗ + κe, z∗〉 = 〈z, z∗〉 ≥ 0. (8.10) Then by (8.9) and (8.8), Az∗ = Pz∗ + Sz∗ = Pz∗ + 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∗) ∈ graT by (8.9). Hence T is maximally monotone. (ii): Let (x∗∗0 , x ∗ 0) ∈ X∗∗ ×X∗. Then we have (x∗∗0 , x ∗ 0) ∈ graT ∗ ⇔ 〈x∗0, Sx∗〉+ 〈x∗, x∗∗0 〉 = 0, ∀x∗ ∈ {e}⊥ ⇔ 〈x∗, x∗∗0 − Sx∗0〉 = 0, ∀x∗ ∈ {e}⊥ ⇔ x∗∗0 − Sx∗0 ∈ ({e}⊥)⊥ = span{e} (by Fact 3.1.1) ⇔ x∗∗0 − Sx∗0 = re, ∃r ∈ R. 188 8.2. Main construction Thus graT ∗ = {(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∗), ∃r ∈ R, z∗ ∈ X∗ ⇔ (z, z∗) = (−Az∗ + [〈z∗, e〉 − r] e, z∗), ∃r ∈ R, z∗ ∈ X∗ (by (8.8)) ⇔ (z, z∗) = (−Az∗, z∗), 〈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∗(domFT )− PX∗(domF ) ) = 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 (FT1F )(x, x ∗) ≥ 〈x, x∗〉 > −∞. (8.13) Then by (8.12), (8.13) and Fact 8.1.3, (FT1F ) ∗(v∗0 , v0) = min x∗∗∈X∗∗ F ∗T (v ∗ 0 , x ∗∗) + F ∗(v∗0 , v0 − x∗∗) ≤ F ∗T (v∗0 , A∗v∗0) + F ∗(v∗0 , v0 −A∗v∗0) = 0 + F ∗(v∗0 , v0 −A∗v∗0) (by (iv)) = (f ⊕ f∗)∗(v∗0 , v0 −A∗v∗0) = (f∗ ⊕ f∗∗)(v∗0 , v0 −A∗v∗0) = f∗(v∗0) + f ∗∗(v0 −A∗v∗0) < 〈v∗0 , v0〉 (by 8.7). (8.14) Hence FA1F is not a BC–function. (viii): By the assumption, there exists x∗0 ∈ dom f∗∗ ◦ A∗|X∗ such that 〈e, x∗0〉 6= 0. Let ε0 = 〈e,x ∗ 0 〉2 2 . By [92, Theorem 2.4.4(iii)]), there exists y∗∗∗0 ∈ ∂ε0f∗∗(A∗x∗0). By [92, Theorem 2.4.2(ii)]), f∗∗(A∗x∗0) + f ∗∗∗(y∗∗∗0 ) ≤ 〈A∗x∗0, y∗∗∗0 〉+ 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 y∗0 ∈ X∗ such that f∗∗(A∗x∗0) + f ∗(y∗0) < 〈A∗x∗0, y∗0〉+ 2ε0. (8.16) Let z∗0 = y ∗ 0 + x ∗ 0. Then by (8.16), we have f∗∗(A∗x∗0) + f ∗(z∗0 − x∗0) < 〈A∗x∗0, z∗0 − x∗0〉+ 2ε0 = 〈A∗x∗0, z∗0〉 − 〈A∗x∗0, x∗0〉+ 2ε0 = 〈A∗x∗0, z∗0〉 − 〈x∗0, Ax∗0〉+ 2ε0 = 〈A∗x∗0, z∗0〉 − 2ε0 + 2ε0 = 〈A∗x∗0, z∗0〉. (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 (FT2F )(x, x ∗) ≥ 〈x, x∗〉 > −∞. (8.18) Then by (8.18), (iv) and Fact 7.1.5, (FT2F ) ∗(z∗0 , A ∗x∗0) 191 8.3. Examples and applications = min y∗∈X∗ F ∗T (y ∗, A∗x∗0) + F ∗(z∗0 − y∗, A∗x∗0) ≤ F ∗T (x∗0, A∗x∗0) + F ∗(z∗0 − x∗0, A∗x∗0) = 0 + F ∗(z∗ − x∗0, A∗x∗0) (by (iv)) = (f ⊕ f∗)∗(z∗0 − x∗0, A∗x∗0) = f∗(z∗0 − x∗0) + f∗∗(A∗x∗0) < 〈z∗0 , A∗x∗0〉 (by (8.17)). (8.19) Let F0 : X ×X∗ → ]−∞,+∞] be defined by (x, x∗) 7→ 〈x, x∗〉+ ιgra(T+∂f)(x, x∗). (8.20) Clearly, FT2F ≤ F0 on X ×X∗ and thus (FT2F )∗ ≥ F ∗0 on X∗ ×X∗∗. By (8.19), F ∗0 (z ∗ 0 , A ∗x∗0) < 〈z∗0 , 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 ‖ · ‖∞ so that X∗ = `1(N) with norm ‖ · ‖1, and X∗∗ = `∞(N) with norm ‖ · ‖∗. Let α = (αn)n∈N ∈ `∞(N) with lim supαn 6= 0, and let Aα : `1(N)→ `∞(N) be defined by (Aαx ∗)n = α2nx ∗ n + 2 ∑ i>n αnαix ∗ i , ∀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 graTα = { (−Sαx∗, x∗) ∣∣ x∗ ∈ X∗, 〈α, x∗〉 = 0} = { (−Aαx∗, x∗) ∣∣ x∗ ∈ X∗, 〈α, x∗〉 = 0} = {( (− ∑ i>n αnαix ∗ i + ∑ i<n αnαix ∗ i )n∈N, x ∗)∣∣∣x∗ ∈ X∗, 〈α, x∗〉 = 0}. (8.21) Then the following hold. (i) 〈Aαx∗, 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) FTα1(‖ · ‖ ⊕ ιBX∗ ) is not a BC–function. (iv) Tα + ∂‖ · ‖ is a maximally monotone operator that is not of type (D). (v) If 1√ 2 < ‖α‖∗ ≤ 1, then FTα1(12‖ · ‖2 ⊕ 12‖ · ‖21) 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)→ `∞(N) be Gossez’s operator [50] defined by ( G(x∗) ) n∈N = ∑ i>n x∗i − ∑ i<n x∗i , ∀(x∗n)n∈N ∈ `1(N). Then Te : c0 ⇒ ` 1(N) as defined by graTe = {(−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 ∈ `∞(N), Aα is linear and continuous and ranAα ⊆ c0 ⊆ `∞(N). (i): We have 〈Aαx∗, x∗〉 = ∑ n x∗n(α 2 nx ∗ n + 2 ∑ i>n αnαix ∗ i ) = ∑ n α2nx ∗ n 2 + 2 ∑ n ∑ i>n αnαix ∗ nx ∗ i = ∑ n α2nx ∗ n 2 + ∑ n 6=i αnαix ∗ nx ∗ i = ( ∑ n αnx ∗ n) 2 = 〈α, x∗〉2, ∀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 Pαx ∗ = 〈α, x∗〉α (for every x∗ ∈ `1(N)). Thus, the skew part Sα of Aα is (Sαx ∗)n∈N = (Aαx∗)n∈N − (Pαx∗)n∈N = ( α2nx ∗ n + 2 ∑ i>n αnαix ∗ i − ∑ i αnαix ∗ i ) n∈N = (∑ i>n αnαix ∗ i − ∑ i<n αnαix ∗ i ) n∈N . (8.23) 194 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 6= 0, there exists i0 ∈ N such that αi0 6= 0. Let ei0 = (0, . . . , 0, 1, 0, . . .), i.e., the i0th component is 1 and the others are 0. Then by (8.23), we have Sαei0 = αi0(α1, . . . , αi0−1, 0,−αi0+1,−αi0+2, . . .). (8.24) Then A∗αei0 = Pαei0 − Sαei0 = αi0(0, . . . , 0, αi0 , 2αi0+1, 2αi0+2, . . .). (8.25) Now set v∗0 = ei0 and v0 = 3‖α‖2∗ei0 . Thus by (8.25), v0 −A∗αv∗0 = 3‖α‖2∗ei0 −A∗αei0 = (0, . . . , 0, 3‖α‖2∗ − α2i0 ,−2αi0αi0+1,−2αi0αi0+2, . . .) (8.26) We have f∗(v∗0) + f ∗∗(v0 −A∗αei0) = ιBX∗ (ei0) + ‖v0 −A∗αei0‖∗ = ∥∥∥3‖α‖∗ei0 −A∗αei0∥∥∥∗ < 3‖α‖2∗ (by (8.26)) = 〈v0, v∗0〉. 195 8.3. Examples and applications Hence by Theorem 8.2.2 (vii), FTα1(‖ · ‖ ⊕ ιBX∗ ) is not a BC–function. (iv): Let f = ‖ · ‖ on X. Since dom f∗∗ = X∗∗, ∅ 6= dom f∗∗ ◦ A∗α|X∗ " {e}⊥. Then apply Theorem 8.2.2(v)&(viii) directly. (v): Let f = 12‖ · ‖2 on X = c0. Then f∗ = 12‖ · ‖21 and f∗∗ = 12‖ · ‖2∗. By 1√ 2 < ‖α‖∗ ≤ 1, take |αi0 |2 > 12 . Let ei0 be defined as in the proof of (iii). Then set v∗1 = 1 2ei0 and v1 = ( 1 + 12α 2 i0 ) ei0 . By (8.25), we have v1 −A∗αv∗1 = (0, . . . , 0, 1,−αi0αi0+1,−αi0αi0+2, . . .) (8.27) Since |αi0αj | ≤ ‖α‖2∗ ≤ 1, ∀j ∈ N, then ‖v1 −A∗αv∗1‖ ≤ 1. (8.28) We have f∗(v∗1) + f ∗∗(v1 −A∗αv∗1) = 12‖v∗1‖21 + 12‖v1 −A∗αv∗1‖2∗ ≤ 18 + 12 (by (8.28)) < α2i0 4 + 1 2 (by α 2 i0 > 1 2 ) = 〈v∗1 , v1〉. Hence by Theorem 8.2.2(vii), FTα1( 1 2‖ · ‖2 ⊕ 12‖ · ‖2∗) is not a BC–function. (vi): Let λ > 0 and f = λ2‖ · ‖2 on X = c0. Then f∗∗ = λ2‖ · ‖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, Proposi- tion 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 Exam- ple 8.3.1(viii) was established by Voisei and Zălinescu 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. Ex- ample 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 Zălinescu’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 symmet- ric 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 subdiffer- ential 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 set- valued 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)|domA is a singleton. As we shall see in Section 9.1, the following decomposition is less restric- tive. 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 previ- ously 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 para- monotone, 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 domA 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′ : domA− {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 domA′, we have that ∂f and R 201 9.1. Decompositions are single-valued on domA′ and that R = A′ − ∂f on domA′. 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 domA′, it follows that domA′ ⊆ D′ ⊆ argmin f . Since ∂f is single-valued on domA′, ∂f(x) = {0}, ∀x ∈ domA′. 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 single- 202 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. (i) ∂f + IdomA : x 7→ ∂f(x), if x ∈ domA; ∅, otherwise is a monotone linear relation. (ii) domA ⊆ dom ∂f ⊆ dom f ⊆ (A0)⊥. (iii) If A is maximally monotone, then domA ⊆ dom ∂f ⊆ dom f ⊆ domA. (iv) If A is maximally monotone and domA is closed, then dom∂f = domA = dom f . 203 9.1. Decompositions Proof. (i): Indeed, on domA, we see that ∂f = A − S is the difference of two linear relations. (ii): Clearly domA ⊆ 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 6∈ (A0)⊥, then σA0(x) = +∞ since A0 is a linear subspace, so f(x) = +∞, ∀x 6∈ (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 domA ⊆ domA∗. 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 domA. Proof. From Proposition 3.2.10(i), A+ is monotone and qA+ = qA, using Proposition 3.2.10(ii), graA+ ⊆ gra ∂qA+ = gra ∂qA. Let S : domA → X∗ be a linear selection ofA◦ (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 graA = gra(A+ + S) ⊆ gra(∂qA + S). Since A is maximally mono- tone, A = ∂qA+S, which is the announced Borwein-Wiersma decomposition. Moreover, ∂qA = A− S = A+ on domA. 204 9.1. Decompositions Corollary 9.1.10 Let A : X ⇒ X∗ be a maximally monotone linear rela- tion 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- Wiersma decomposable with A−1 = ∂q∗A + 0. Proof. Using Proposition 3.2.11, we obtain A = A∗|X . Hence, Theo- rem 9.1.9 applies; in fact, A = ∂qA. If X is reflexive, then we have A−1 = ∂qA∗ = ∂q∗A by [92, Theorem 2.4.4(iv) and Theorem 2.3.1(iv)]. From Proposition 3.1.3(iv), we have A−1 = (A∗)−1 = (A−1)∗. Then A−1 = ∂qA−1 . Hence A−1 = ∂qA−1 = ∂q∗A. Corollary 9.1.11 Let A : X ⇒ X∗ be a maximally monotone linear rela- tion such that domA 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 domA∗|X = domA. By Proposi- tion 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 Borwein- Wiersma decomposition A = ∂qA+S. Hence domA ⊆ dom ∂qA ⊆ dom qA ⊆ domA = domA. In turn, since domA = domA+ and qA = qA+, this implies that domA+ = dom ∂qA+ = dom qA+ . In view of Proposition 3.2.10(i)&(ii), qA+ = qA+ and graA+ ⊆ gra ∂qA+. By Theorem 9.1.9, A+ = ∂qA on domA. Since domA = domA+ = 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 ∂ιdomA + S. Proof. Clearly, S is skew. Proposition 3.1.3(ii) and Proposition 3.2.2(iii) imply that A = A0 + S = (domA)⊥ + S = ∂ιdomA + S, as announced. Alternatively, by [80, Lemma 2.2], domA ⊆ domA∗ and now apply Theo- rem 9.1.9. Under a mild constraint qualification, the sum of two Borwein-Wiersma decomposable operators is also Borwein-Wiersma decomposable and the de- composition 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 Borwein- Wiersma decomposable via A1 = ∂f1 + S1 and A2 = ∂f2 + S2, respectively. Suppose that domA1−domA2 is closed. Then A1+A2 is Borwein-Wiersma decomposable via A1 +A2 = ∂(f1 + f2) + (S1 + S2). Proof. By Lemma 9.1.8(iii), domA1 ⊆ dom f1 ⊆ domA1 and domA2 ⊆ dom f2 ⊆ domA2. Hence domA1 − domA2 ⊆ dom f1 − dom f2 ⊆ domA1 − domA2 ⊆ domA1 − domA2 = domA1− domA2. Thus, dom f1− dom f2 = domA1 − domA2 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 domA. 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 domA is closed, the Brøndsted- Rockafellar Theorem (see [74, Theorem 18.6]) implies that dom f = Y = domA. By Proposition 3.2.2(iv), domA∗|X = domA. Let x ∈ Y and consider the directional derivative g = f ′(x; ·), i.e., g : X → [−∞,+∞] : y 7→ lim t↓0 f(x+ ty)− f(x) t . By [92, Theorem 2.1.14], dom g = ⋃ r≥0 r · (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〉 s = lim s↓0 f ′(x+ sz; y)− f ′(x; y) s (9.10) = lim s↓0 lim t↓0 (f(x+ sz + ty)− f(x+ sz) st − f(x+ ty)− f(x) st ) . Set h : R→ R : s 7→ 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) s − f(x+ ty)− f(x) s = h(s) s − h(0) s = h′(rs,t) (9.11) = f ′(x+ rs,tz + ty; z)− f ′(x+ rs,tz; 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 domA. Set B = (A+ IZ) + Z⊥. Then B is maximally monotone and domB = Z. 208 9.2. Uniqueness results Proof. Since Z ⊆ domA and B = A+ ∂ιZ it is clear that B is a monotone linear relation with domB = Z. By Proposition 3.2.2 (i), we have Z⊥ ⊆ B0 = A0 + Z⊥ ⊆ (domA)⊥ + Z⊥ ⊆ Z⊥ + Z⊥ = Z⊥. Hence B0 = Z⊥ = (domB)⊥. 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∗) ∈ graA. Set Z = span{x, y}. Let B : X ⇒ X∗ be defined as in Lemma 9.2.2. Clearly, graB ⊆ gra ∂(f+ιZ). In view of the maximal monotonicity of B, we see that B = ∂(f+ ιZ). Since domB = 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 domA ⊆ domA∗. 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 domA ⊆ domS, and Theorem 9.2.3 implies that (A− S) + IdomA is symmetric. Let x and y be in domA. 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, domA = dom(A − 2S) ⊆ domA∗. Remark 9.2.5 We can now derive part of the conclusion of Proposition 9.1.13 differently as follows. Since domA1−domA2 is closed, Voisei proved in [83] (see Theorem 7.2.2 or [74, Theorem 46.3]) that A1+A2 is maximally mono- tone; moreover, Fact 7.1.6 yields (A1+A2) ∗ = A∗1+A ∗ 2. Using Lemma 9.2.4, we thus obtain dom(A1 + A2) = domA1 ∩ domA2 ⊆ domA∗1 ∩ domA∗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 Borwein- Wiersma decomposable. Then on domA, 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 = domA. 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) S1x−S2x ∈ ∂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〉 = 〈S1x− S2x, y〉 (9.15) 211 9.2. Uniqueness results = −〈S1y − S2y, 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⊥ = (domA)⊥ = 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 S1x ∈ 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 graS = {( (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 subdiffer- ential 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 domA. (ii) If domA 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, domA ⊆ domA∗. In turn, Theorem 9.1.9 yields A = ∂qA + S. Let {x, y} ⊂ domA. By Theorem 9.2.8, ∂f + IdomA = ∂qA + IdomA. Now set Z = span{x, y}, apply Lemma 9.2.2 to the monotone linear relation ∂f + IdomA = ∂qA + IdomA, and let B be as in Lemma 9.2.2. Note that graB = gra(∂qA + ∂ιZ) ⊆ gra ∂(qA+ ιZ) and that graB = 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 ∈ (domA)r {x}, we see that f = qA + α on domA (9.16) and (i) is thus established. Now assume in addition that domA is closed. Applying Lemma 9.1.8(iv) with both ∂f and ∂qA, we obtain dom qA = dom ∂qA = domA = 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) domA ⊆ domA∗. (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 de- composable if and only if domA = domA∗. 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 Example 9.3.3 Suppose that X is the Hilbert space `2(N), and set S : domS → X : y 7→ ( 1 2yn + ∑ i<n yi ) n∈N , (9.17) with domS = { y = (yn)n∈N ∈ X ∣∣∣∣ ∑ i≥1 yi = 0, (∑ i≤n yi ) n∈N ∈ X } . Then S∗ : domS∗ → X : y 7→ ( 1 2yn + ∑ i>n yi ) n∈N (9.18) where domS∗ = { y = (yn)n∈N ∈ X ∣∣∣∣ (∑ i>n yi ) 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, Propo- sition 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) domS is dense in `2(N), and domS $ domS∗. (iv) S∗ = −S on domS. 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 The- orem 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 V : X → X : x 7→ V x, where V x : [0, 1]→ R : t 7→ ∫ t 0 x, (9.19) and set A = V −1. Then V ∗ : X → X : x 7→ V ∗x, where V ∗x : [0, 1] → R : t 7→ ∫ 1 t x, and the following hold. (i) We have domA = { x ∈ X ∣∣ x is absolutely continuous, x(0) = 0, and x′ ∈ X} and A : domA→ X : x 7→ x′. (ii) We have domA∗ = { x ∈ X ∣∣ x is absolutely continuous, x(1) = 0, 216 9.3. Characterizations and examples and x′ ∈ X} and A∗ : domA∗ → X : x 7→ −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) domA 6⊆ domA∗, and domA∗ 6⊆ domA. (vii) Y = domA ∩ domA∗ 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 = P ∗ Y . Definition 9.4.1 Let A : X ⇒ X be a maximally monotone linear relation. We define QA by QA : domA→ X : x 7→ PAxx. Note that QA is monotone and a single-valued selection of A because (∀x ∈ domA) 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 ∈ domA) QAx = P(A0)⊥(Ax), and QAx ∈ Ax. (ii) QA is monotone and linear. (iii) A = QA +A0. Proof. Let x ∈ domA = domQA and let x∗ ∈ Ax. Using Proposition 3.1.3(ii), we see that QAx = PAxx = Px∗+A0x = x ∗ + PA0(x− x∗) = x∗ + PA0x− PA0x∗ = PA0x+ 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 domA, and let α and β be in R. If α = β = 0, then, by Proposi- tion 3.1.3(i), we have QA(αx + βy) = QA0 = PA00 = 0 = αQAx + βQAy. Now assume that α 6= 0 or β 6= 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) = αQAx+ βQAy. 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 = ∂ιdomA +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 domA is closed. Set B = PdomAQAPdomA and f = qB + ιdomA. 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 = ∂ιdomA +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 selec- tion of A. Hence, B : X → X is linear; moreover, (∀x ∈ X) 〈x,Bx〉 = 〈x, PdomAQAPdomAx〉 = 〈PdomAx, QAPdomAx〉 ≥ 0. Altogether, B : X → X is linear and monotone. By Corollary 3.2.3, B is continuous and maxi- mally 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) (QAPdomA)x ∈ (A0)⊥ = domA = domA. Hence, (∀x ∈ domA) Bx = (PdomAQAPdomA)x = QAx ∈ Ax. Thus, B + IdomA = QA. In view of Proposition 9.4.2(iii) and Proposition 3.2.2(iii), we now obtain A = B + IdomA +A0 = B + ∂ιdomA. (iv): It follows from (iii) and (9.2) that A = B + ∂ιdomA = ∇qB + ∂ιdomA +B◦ = ∂(qB + ιdomA) +B◦ = ∂f +B◦. Proposition 9.4.5 Let A : X ⇒ X be such that domA is a closed subspace of X. Then A is a maximally monotone linear relation ⇔ A = ∂ιdomA+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 mono- tone. Using Rockafellar’s sum theorem [66] or Theorem 5.3.1, we conclude that ∂ιdomA +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 max- imally 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, this fails in infinite-dimensional settings. We presented characterizations of Borwein- Wiersma 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 correspond- ing 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 im- portant characterization of maximal monotonicities of monotone relations. The original proof [30] is based on the application of Zorn’s Lemma by con- structing 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 Brezis- Browder Theorem in a general Banach space: Let A : X ⇒ X∗ be a monotone linear relation such that graA 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 con- crete examples for the characterization of decomposable monotone linear relations. Chapter 4: A direction for future work in this chapter is to write com- puter 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 domA ∩ intC 6= ∅. 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 domA ∩ int domB 6= ∅. Assume that domA ∩ domB ⊆ domB. Then A+B is maximally monotone. Chapter 6: Our first main result (Theorem 6.2.1) in this chapter is ob- tained 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 [ domA− ⋃ λ>0 λC ] is a closed subspace of X. Is it necessarily true that FA+NC = FA2FNC? Chapter 8: The idea of the construction of the operator A in (Theo- rem 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 Zălinescu’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 mono- tone 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. Zălinescu, Convex Analysis in General Vector Spaces, World Scien- tific Publishing, 2002. [93] E. Zeidler, Nonlinear Functional Analysis and its Application, Vol II/B Nonlinear Monotone Operators, Springer-Verlag, New York-Berlin- Heidelberg, 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 Student:−LinearAlgebra 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 , t h i ckn e s s = 2 , co lou r = blue ) 239 Appendix A. Maple code The following is the Maple code used to verify the calculations for Exam- ple 4.5.2 on G̃2 >r e s t a r t : Loading Student:−LinearAlgebra >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. Transpose (B)+B. Transpose (A) >Eigenvalues (T) >Eigenvec tor s (T) >Idlam :=[[[−1+ sq 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:=Transpose (N) . Idlam .N >e v a l f ( E igenvalues (M) ) >NullSpace ( ‘< |> ‘( Transpose (N) . Transpose (V) .A, Transpose (N) . Transpose (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 Exam- ple 4.5.3 on G̃1, G̃2, E1 and E2. >r e s t a r t : Loading Student:−L in ea rA l g eb r avo 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. Transpose (B)+B. Transpose (A) >Eigenvec tor s (K1) >Idlam := Matrix (3 , 3 , {(1 , 1) = 13+sq r t (201) , (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 (201)} ) >V:= Matrix (3 , 3 , {(1 , 1) = 20/(1+ sq r t ( 201 ) ) , (1 , 2) = 0 , (1 , 3) = 20/(1− s q r t ( 201 ) ) , (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 ( 201 ) ) , (2 , 2) = 1 , (2 , 3) = 1}) >L :=NullSpace ( ‘< |> ‘(V g .A, V g .B) ) >C0 := Matrix (2 , 2 , {(1 , 1) = −(−21+sq r t (201))/(−2+2∗ s q r t ( 201 ) ) , (1 , 2) = −(−107+7∗ s q r t (201))/(−2+2∗ s q r t ( 201 ) ) , (2 , 1) = (−23+3∗ s q r t (201))/(−2+2∗ s q r t ( 201 ) ) , (2 , 2) = (−117+17∗ s q r t (201))/(−2+2∗ s q r t ( 201 ) )} ) 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 := Transpose (N) . Idlam .N >e v a l f ( E igenvalues (M) ) >NullSpace ( ‘< |> ‘( Transpose (N) . Transpose (V) .A, Transpose (N) . Transpose (V) .B) ) >C1 := Matrix (2 , 2 , {(1 , 1) = −9/20+(1/30)∗ s q r t (201) , (1 , 2) = −13/4+(1/6)∗ s q r t (201) , (2 , 1) = 29/20−(1/30)∗ s q r t (201) , (2 , 2) = 33/4−(1/6)∗ s q r t (201)} ) >t i l d e {G 2}:= 1/C1 >vec := Vector (3 , {(1) = 0 , (2 ) = 0 , (3 ) = 0}) >LinearSolve ( ‘< |> ‘(A, B, vec ) , f r e e = t ) 242 Index ε–subdifferential operator, 8, 141 adjoint, 5, 27, 28, 147, 153 Asplund decomposition, 200, 203 Attouch & Brezis’ Theorem, 15 BC–function, 4, 181, 182, 187, 193 Borwein’s Theorem, 141 Borwein-Wiersma decomposition, 199, 211, 212, 214 boundary, 7 Brezis & Browder’s Theorem, 27 Censor, Iusem & Zenios’ Theorem, 200 closed unit ball, 8 constraint qualification, 99 convex hull, 7 Crouzeix & Ocaña-Anaya’s char- acterizations, 70 distance function, 7 domain, 5 duality mapping, 121 Fenchel conjugate, 7 Fitzpatrick function, 6, 101 Fitzpatrick, Phelps & Veronas’ The- orem, 103 graph, 5 identity mapping, 167 indicator function, 7 indicator mapping, 7 inf-convolution, 8, 15 interior, 7 inverse operator, 5 irreducible, 199 isometric, 182 isometry, 182 isomorphism into, 182 linear relation, 6 lower semicontinuous hull, 7 243 Index maximally monotone, 6 maximally skew, 40 maximally skew extension, 40 monotone, 6 monotonically related to, 6 norm closure, 7 open unit ball, 8 paramonotone, 200 partial inf-convolution, 156, 159, 182 range, 5 representative, 140, 142 right and left shift operator, 39 Rockafellar’s Theorems, 99, 100 set-valued operator, 5 Simons & Veronas’ Theorem, 103 Simons & Zălinescu’s Theorems, 28, 159, 182 Simons’ Theorems, 101–103, 141, 183 Simons, Marques Alves & Svaiter’s Theorem, 142 skew, 19 skew part, 19 subdifferential operator, 8, 100, 102, 103, 141 sum operator, 99 sum problem, 99, 115, 122 symmetric, 19 symmetric part, 19 type (D), 140, 146, 153, 184 type (FPV), 6, 103, 115, 133 type Fitzpatrick-Phelps (FP), 140, 141, 143, 146, 153 type Fitzpatrick-Phelps-Veronas, 6 type negative infimum (NI), 140, 143, 146, 147, 153 Voisei’s Theorem, 103 Volterra integration operator, 43 weak closure, 7 244
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