Science, Irving K. Barber Faculty of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Song, Shuang
Master of Science - MSc
University of British Columbia
Averaged operators are important in convex analysis and optimization algorithms. In the first part of this thesis, we propose classifications of averaged operators, firmly nonexpansive operators, and proximal operators by using the Bauschke-Bendit-Moursi modulus of averagedness. We show that if an averaged operator has modulus of averagedness less than 1/2, then it is a bi-Lipschitz homeomorphism. Amazingly the proximity operator of a convex function has its modulus of averagedness less than 1/2 if and only if the function is Lipschitz smooth. Some results on the averagedness of operator compositions are obtained. Explicit formulae for calculating the modulus of averagedness of resolvents and proximity operators in terms of various values associated with the maximally monotone operator or subdifferential are also given. Examples are provided to illustrate our results. In the second part of this thesis, we focus on a natural generalization of averaged operators and monotonicity. We extend the definition of modulus of averagedness and show how it behaves for linear operators. As an application, explicit formulae are obtained with respect to angle between subspaces. We will also present various profound nonlinear results about this generalization of averaged operator and monotonicity. All the results can apply to averaged operators. Examples are provided to illustrate our results.
On Bauschke-Bendit-Moursi modulusof averagedness and generalizedmonotone operatorsbyShuang SongB.Sc., Sichuan University, 2023A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)December 2024© Shuang Song, 2024The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:On Bauschke-Bendit-Moursi modulus of averagedness and gen-eralized monotone operatorssubmitted by Shuang Song in partial fulfilment of the requirements of thedegree of Master of ScienceDr. Shawn X. Wang, Irving K. Barber Faculty of ScienceSupervisorDr. Heinz Bauschke, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Amy Wiebe, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Stephen Karrer O’Leary, School of EngineeringUniversity ExamineriiAbstractAveraged operators are important in convex analysis and optimizationalgorithms. In the first part of this thesis, we propose classifications of av-eraged operators, firmly nonexpansive operators, and proximal operators byusing the Bauschke-Bendit-Moursi modulus of averagedness. We show thatif an averaged operator has modulus of averagedness less than 1/2, then it isa bi-Lipschitz homeomorphism. Amazingly the proximity operator of a con-vex function has its modulus of averagedness less than 1/2 if and only if thefunction is Lipschitz smooth. Some results on the averagedness of operatorcompositions are obtained. Explicit formulae for calculating the modulusof averagedness of resolvents and proximity operators in terms of variousvalues associated with the maximally monotone operator or subdifferentialare also given. Examples are provided to illustrate our results.In the second part of this thesis, we focus on a natural generalizationof averaged operators and monotonicity. We extend the definition of mod-ulus of averagedness and show how it behaves for linear operators. As anapplication, explicit formulae are obtained with respect to angle betweensubspaces. We will also present various profound nonlinear results aboutthis generalization of averaged operator and monotonicity. All the resultscan apply to averaged operators. Examples are provided to illustrate ourresults.iiiLay SummaryThis research concerns averaged operators, which are frequently usedin modern algorithms. We investigate a modulus associated with an av-eraged operator which was very recently proposed. We classify averagedoperators using this modulus, and prove that several subclasses of averagedoperator have very nice properties. We later focus on a generalization ofaveraged operator and propose the generalized modulus concept. Then weshow how to calculate the modulus when the operators are linear. Moreover,we abstractly deduce several nonlinear properties of the modulus. Monotoneoperators and its generalization are investigated as well.ivPrefaceThis thesis contains two parts. Part one of this thesis, namely fromChapter 1 to Chapter 5, is based on the manuscript [SW24] co-authored withDr. Shawn X. Wang. The manuscript has been submitted for publicationto Canadian Journal of Mathematics.Part two of this thesis, namely from Chapter 6 to Chapter 8, includesthe most recent and unpublished work done by me.Generative artificial intelligence (AI) was not used for any aspects of thiswork.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . ixChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2: Bijective theorem . . . . . . . . . . . . . . . . . . . 42.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Injective lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Bijective theorem . . . . . . . . . . . . . . . . . . . . . . . . . 7Chapter 3: Application . . . . . . . . . . . . . . . . . . . . . . . 93.1 Compact operator . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Operator composition . . . . . . . . . . . . . . . . . . . . . . 93.3 Limiting operator . . . . . . . . . . . . . . . . . . . . . . . . . 12Chapter 4: Resolvent . . . . . . . . . . . . . . . . . . . . . . . . 164.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Characterization of normally nonexpansive operators . . . . . 184.3 Cocoercive value . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Monotone value . . . . . . . . . . . . . . . . . . . . . . . . . . 23Chapter 5: Proximal operator . . . . . . . . . . . . . . . . . . . 265.1 Characterization of normal proximal operators . . . . . . . . 26viTABLE OF CONTENTS5.2 Lipschitz value . . . . . . . . . . . . . . . . . . . . . . . . . . 27Chapter 6: Conical averagedness and generalized monotonicity 306.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2 Properties of BBM modulus . . . . . . . . . . . . . . . . . . . 316.3 Monotone value and comonotone value . . . . . . . . . . . . . 34Chapter 7: Linear world . . . . . . . . . . . . . . . . . . . . . . . 417.1 Linear world . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 Singular case . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.3 Angle between subspaces . . . . . . . . . . . . . . . . . . . . . 457.4 BBM modulus as a function . . . . . . . . . . . . . . . . . . . 497.5 On Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Chapter 8: Nonlinear results . . . . . . . . . . . . . . . . . . . . 538.1 Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . 538.2 Continuity analysis . . . . . . . . . . . . . . . . . . . . . . . . 588.3 More on the comonotone value . . . . . . . . . . . . . . . . . 618.4 On the displacement operator . . . . . . . . . . . . . . . . . . 668.5 Lipschitz moduli space . . . . . . . . . . . . . . . . . . . . . . 71Chapter 9: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 75Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Appendix A: Alternative proofs . . . . . . . . . . . . . . . . . . . . 80A.1 A topological proof of Lemma 2.11 . . . . . . . . . . . . . . . 80A.2 A proof of Lemma 2.11 when T is linear . . . . . . . . . . . . 81A.3 Direct proofs of Example 5.15 (iii) . . . . . . . . . . . . . . . 81viiList of FiguresFigure 8.1 Bounds comparison for α = 12 . . . . . . . . . . . . . 56viiiAcknowledgementsI would like to express my sincere gratitude to my supervisor Dr. ShawnX. Wang for his consistent guidance, encouragement, and expertise providedin this research journey.I would also like to thank Dr. Heinz Bauschke for his inspiring lectures.I am grateful to the committee members for the time they spent readingthis thesis and the helpful feedback they provided.ixChapter 1IntroductionThroughout, without specific statement, we assume thatX is a real Hilbert space with inner product ⟨·, ·⟩ : X ×X → R,and induced norm ∥ · ∥. Let Id denote the identity operator on X.Definition 1.1. Let T : X → X. We say T is nonexpansive1 if(∀x ∈ X)(∀y ∈ X) ∥Tx− Ty∥ ⩽ ∥x− y∥.Definition 1.2. Let T : X → X be nonexpansive. We say T is k-averaged2if T can be represented as T = (1 − k)Id + kN , where N is nonexpansiveand k ∈ [0, 1].Averaged operators, firstly introduced in [BBR78], are important in anal-ysis and optimization. From the definition we have Id is the only 0-averagedoperator. When k ∈ (0, 1], various characterizations of k-averagedness areavailable including(∀x ∈ X)(∀y ∈ X) ∥Tx−Ty∥2 ⩽ ∥x−y∥2− 1− kk∥(Id−T )x− (Id−T )y∥2(1.1)and(∀x ∈ X)(∀y ∈ X) ∥Tx−Ty∥2+(1−2k)∥x−y∥2 ⩽ 2(1−k)⟨x−y, Tx−Ty⟩,(1.2)see [BC17, Proposition 4.35]. From (1.1) and the fact that Id is the only0-averaged operator, we can deduce that if an operator is k0-averaged, thenit is k-averaged if k0 ⩽ k. This motivates the following definition, which wasproposed by [BBM23].1We assume T has full domain for convenience throughout this thesis while one cangeneralize it to be proper subsets of X.2Usually, one excludes the cases k = 0 and k = 1 in the study of averaged operators,but it is very convenient in this thesis to allow for this case.1Definition 1.3. (Bauschke-Bendit-Moursi modulus of averagedness)Let T : X → X be nonexpansive. Then the modulus of averagedness of Tis defined byk(T ) := inf{k ∈ [0, 1] | T is k-averaged }.We call it as the BBM modulus of averagedness.Recall that we say T : X → X is firmly nonexpansive if T is 12 -averaged.Concerning the definition above, this is equivalent to say k(T ) ⩽ 12 . Based onthis, we define the following, which classifies firmly nonexpansive operatorsusing BBM modulus of averagedness.Definition 1.4. (normal and special nonexpansiveness) Let T : X →X. We say T is normally (firmly) nonexpansive if k(T ) < 12 and T is specially(firmly) nonexpansive if k(T ) = 12 .k ⩽ 1k = 12k < 12Let Γ0(X) denote the set of all proper lower semicontinuous convex func-tions from X to (−∞,+∞] and let f ∈ Γ0(X). Recall the proximal operatorof f is defined byPf (x) = argminw∈X{f(w) +12∥w − x∥2}.It is well-known that Pf is firmly nonexpansive, i.e., k(Pf ) ⩽ 12 . Somenatural questions arise: when is Pf normally (or specially) nonexpansive?Given a specific f ∈ Γ0(X), how can we calculate k(Pf )? In [BBM23], theseproblems are essentially solved in linear cases, or, in the smooth case whenX = R.2One of the goals of this thesis is to solve the above problems in nonlinearand nonsmooth cases in a general Hilbert space. Moreover, we aim to de-termine what properties the normally (or specially) nonexpansive operatorspossess and to characterize them using monotone operator theory.The rest of this thesis is organized as follows. In Chapter 2 we willexplore properties of the BBM modulus of averagedness, and show thatnormally nonexpansive operators are bijective.In Chapter 3 we will explore applications of these properties on operatorcomposition and connection with compact operators. The roles of Averagedoperators in algorithms are also examined.In Chapter 4 and 5 we will investigate both normal and special nonex-pansiveness of resolvent and proximal operator. Characterizations on theseconcepts are obtained. By defining various new values associated with maxi-mally monotone operator, we will also provide formulae to calculate modulusof averagedness.Furthermore, starting from Chapter 6 we will survey a natural generaliza-tion of averaged operators and monotonicity. Corresponding generalizationsof the modulus and various values are examined and analyzed.Chapter 7 will be about how these generalized concepts behave in linearworld. As an application, we will propose explicit formulae with respect toangle between subspaces. The properties of modulus as a function on matrixspace are also discussed.In Chapter 8 we will present several profound nonlinear results. We willinvestigate how scalar multiplication impacts on the modulus. Then varioususeful corollaries are obtained. We will explore more on comonotone valueand the displacement operator. We will also formally define an operatorspace. Examples are provided to illustrate our results.Chapter 9 gives conclusions and directions for future work on these top-ics.In this thesis, previously established results are appropriately cited, whileany uncited results are, to the best of the author’s knowledge, original con-tributions.3Chapter 2Bijective theoremIn this chapter, we will provide several basic results of modulus of aver-agedness. We will prove that normally nonexpansive operators are bijectiveLipschitz homeomorphisms.The main new results of this chapter are Theorem 2.6 and Theorem 2.13.2.1 PreparationFor any operator T : X → X and any v ∈ X, the operator T + v isdefined by(∀x ∈ X) (T + v)x := Tx+ v.Proposition 2.1. (translation invariance) Let T : X → X be nonexpan-sive and v ∈ X. Then(i) k(T + v) = k(T ).(ii) k(T (·+ v)) = k(T ).Proof. (i) The result follows by combining (T + v)x− (T + v)y = Tx− Tywith the characterization (1.2).(ii) The result follows by combining x − y = (x + v) − (y + v) with thecharacterization (1.2).The following proposition concerns whether the infimum in the modulusof averagedness can be attained. Similar argument appeared in [BBM23].Proposition 2.2. Let T : X → X be nonexpansive. If k(T ) > 0, then T isk(T )-averaged.Proof. Due to k(T ) > 0, we can use characterization (1.2). Both the leftand right hand side of (1.2) are continuous functions in k, thus the resultfollows.It is obvious that k(Id) = 0. However, the converse does not hold. Wehave the following proposition characterizing k(T ) = 0.42.2. INJECTIVE LEMMAProposition 2.3. Let T : X → X be nonexpansive. Thenk(T ) = 0⇔ ∃v ∈ X : T = Id+v.Proof. Suppose ∃v ∈ X : T = Id + v. Obviously k(Id) = 0. Thus byProposition 2.1, k(T ) = k(Id + v) = 0.Suppose k(T ) = 0. Assume that for any v ∈ X: T ̸= Id+v. Then thereexist x0, y0 ∈ X, such that (T − Id)x0 ̸= (T − Id)y0. Thus ∥(T − Id)x0 −(T − Id)y0∥2 > 0. Our assumption implies T ̸= Id, and Id is the only 0-averaged operator, thus there exists a sequence (kn)n∈N in (0, 1], such thatT is kn-averaged and kn → 0.Now characterization (1) implies that for any n ∈ N:∥Tx0 − Ty0∥2 ⩽ ∥x0 − y0∥2 − 1− knkn∥(Id−T )x0 − (Id−T )y0∥2,i.e.,0 ⩽ ∥x0 − y0∥2 − ∥Tx0 − Ty0∥2 + (1− 1kn) ∥(T − Id)x0 − (T − Id)y0∥2 .Note that ∥(T − Id)x0 − (T − Id)y0∥2 > 0. Now let n→∞, then0 ⩽ −∞which is a contradiction.We have the following formula to calculate the modulus of operators onR.Proposition 2.4. [BBM23, Proposition 2.8] Let g : R → R be a differen-tiable function. Then g is nonexpansive if and only if |g′| ≤ 1 in which casek(g) = (1− inf g′(R)) /2.2.2 Injective lemmaIn this section, we prove that normally nonexpansive operators must beinjective.Lemma 2.5. Let T : X → X be normally nonexpansive. Then X is bi-Lipschitz homeomorphic to ranT . In particular, T is injective.52.2. INJECTIVE LEMMAProof. Without loss of generality we assume k(T ) > 0 in view of Proposition2.3. Then T is k(T )-averaged by Proposition 2.2, i.e., for any x, y ∈ X∥Tx− Ty∥2 + (1− 2k(T ))∥x− y∥2 ⩽ 2(1− k(T ))⟨x− y, Tx− Ty⟩.Since k(T ) < 12 , there exists α ∈ (0, 12), such that k(T ) = 12 − α. UsingCauchy-Schwarz, we have∥Tx− Ty∥2 + 2α∥x− y∥2 ⩽ (1 + 2α)⟨x− y, Tx− Ty⟩∥Tx− Ty∥2 + 2α∥x− y∥2 ⩽ (1 + 2α) (∥x− y∥ ∥Tx− Ty∥)2α(∥x− y∥2 − ∥x− y∥ ∥Tx− Ty∥) ⩽ ∥x− y∥ ∥Tx− Ty∥ − ∥Tx− Ty∥22α∥x− y∥ (∥x− y∥ − ∥Tx− Ty∥) ⩽ ∥Tx− Ty∥ (∥x− y∥ − ∥Tx− Ty∥) .If ∥x− y∥ − ∥Tx− Ty∥ ≠ 0, then 2α∥x− y∥ ⩽ ∥Tx− Ty∥.If ∥x − y∥ − ∥Tx − Ty∥ = 0, then ∥Tx− Ty∥ = ∥x − y∥ ⩾ 2α∥x − y∥since 2α < 1.Combining this with ∥Tx− Ty∥ ⩽ ∥x− y∥, we have2α∥x− y∥ ⩽ ∥Tx− Ty∥ ⩽ ∥x− y∥.i.e., T is bi-Lipschitz. Thus X is bi-Lipschitz homeomorphic to ranT .It is well-known that convex projections are firmly nonexpansive, i.e.,12 -averaged. But how small could the averaged constant be? The followinganswers this question.Theorem 2.6. (averagedness of projection) Let C be a nonempty closedconvex set in X and C ̸= X. Then PC is specially nonexpansive, i.e.,k(PC) =12 .Proof. It is well-known that PC is firmly nonexpansive, when C is a nonemptyclosed convex set.Now since C ̸= X, let x0 ∈ X\C. Because PC (x0) ∈ C and x0 ∈ X\C,we have PC (x0) ̸= x0. However, PC (x0) = PC (PC (x0)). Thus PC is notinjective. Using the contrapositive of Lemma 2.5 we have PC is not normallynonexpansive. Thus PC is specially nonexpansive.Proposition 2.7. Let M ∈ Rn×n be a square matrix. Suppose M is firmlynonexpansive and det(M) = 0. Then M is specially nonexpansive.62.3. BIJECTIVE THEOREMProof. In finite dimensional linear space, it is well-known that the followingstatements are equivalent for square matrix M : (1) M is not injective.(2) M is not surjective. (3) M is not bijective. (4) det(M) = 0. (5) Mhas eigenvalue 0. Now use the contrapositive of Lemma 2.5, we have Mis not normally nonexpansive. Since M is firmly nonexpansive, the resultfollows.Remark 2.8. (1) There is a useful way to check whether a matrixM is firmlynonexpansive, that is, M is firmly nonexpansive if and only if M +MT −2MTM is positive semidefinite, see [BM23, Proposition 22.6].(2) If det(M) ̸= 0, M may still be specially nonexpansive. For example,letM =13[2 1−1 1].Then using definition, we can verify that k(M) = 12 while det(M) =13 .2.3 Bijective theoremIn this section, we will take use of monotone operator theory to provethat normally nonexpansive operators must be bijective. The full definitionof maximally monotone operator will be given in Chapter 4 and here we willonly use it as an auxiliary tool.Fact 2.9. [BC17, Example 20.30] Let T : X → X be firmly nonexpansive.Then T is maximally monotone.Fact 2.10. (Rockafellar–Vesely)[BC17, Corollary 21.24] Let A : X ⇒ Xbe a maximally monotone operator such thatlim∥x∥→+∞inf ∥Ax∥ = +∞.Then A is surjective.Lemma 2.11. Let T : X → X be normally nonexpansive. Then T issurjective.Proof. By Lemma 2.5, T is bi-Lipschitz since T is normally nonexpansive.Thus there exists ϵ > 0, such that ϵ∥x − y∥ ⩽ ∥Tx − Ty∥. Let y = 0, thenϵ∥x∥ ⩽ ∥Tx− T0∥. Using Triangle inequality, we have∥Tx∥ ⩾ ϵ∥x∥ − ∥T0∥.Thus lim∥x∥→∞ ∥Tx∥ =∞. By Fact 2.9, T is maximally monotone since Tis firmly nonexpansive. Thus using Fact 2.10 we complete the proof.72.3. BIJECTIVE THEOREMRemark 2.12. It is challenging to demonstrate Lemma 2.11 without usingmonotone operator theory. By applying Brouwer’s invariance of domaintheorem (a fundamental result in algebraic topology) we can establish thisresult in Rn. However, attempts to generalize this proof to a general Hilbertspace are unsuccessful.We present in the Appendices alternative proofs of Lemma 2.11 undersome constrained conditions, without using monotone operator theory.Theorem 2.13. (bijective Lipschitz homeomorphism) Let T : X → Xbe normally nonexpansive. Then T is a bi-Lipschitz self-homeomorphism ofX. In particular, T is bijective.Proof. Combine Lemma 2.5 and Lemma 2.11.Taking the contrapositive of Bijective Theorem, we obtain a lower boundfor modulus.Corollary 2.14. Let T : X → X be nonexpansive. Suppose T is not bijec-tive, then k(T ) ⩾ 12 .Corollary 2.15. Let f ∈ Γ0(X). Suppose Pf is not bijective, then Pf isspecially nonexpansive, i.e., k(Pf ) =12 .Proof. By Corollary 2.14, k(Pf ) ⩾ 12 since Pf is not bijective. Since Pf isfirmly nonexpansive, i.e., k(Pf ) ⩽ 12 , we have k(Pf ) =12 .The following example shows that the converse of Theorem 2.13 doesnot hold.Example 2.16. Let X = R. Defineφ(x) :=lnx if x ⩾ e,1ex if −e < x < e,− ln(−x) if x ⩽ −e.Then φ is a specially nonexpansive bijection.Proof. Obviously φ is a bijection. We have φ is differentiable andφ′(x) =1x if x ⩾ e,1e if −e < x < e,− 1x if x ⩽ −e.Thus infx∈R φ′(x) = 0. By Proposition 2.4, we have k(φ) =1−infx∈R φ′(x)2 =12 , i.e., φ is specially nonexpansive.8Chapter 3ApplicationIn this chapter, we will give several applications of Theorem 2.13. Wewill focus on the modulus of composition of two averaged operators. We willdefine a new operator associated with an averaged operator and analyze itsmodulus.The main new results of this chapter are Theorem 3.2, Proposition 3.6and Theorem 3.16.3.1 Compact operatorCompact operators are classic objects in functional analysis, see, e.g.,[Rud91].Definition 3.1. Let T : X → X be linear. Then T is compact if thereexists a neighbourhood U of the origin in X and a compact subset V ⊆ Xsuch that T (U) ⊆ V .Theorem 3.2. There exists a normally nonexpansive compact operator onX if and only if dimX <∞.Proof. Suppose there exists a normally nonexpansive compact operator Ton X. Assume that dimX = ∞. Then by [Rud91, Theorem 4.18] T is notbijective, which contradicts with Theorem 2.13.Suppose dimX < ∞. Then Id is a normally nonexpansive compactoperator.3.2 Operator compositionProposition 3.3. Let T1 and T2 be nonexpansive operators from X to X.Suppose one of the following holds:(i) T1 is not surjective.(ii) T2 is not injective.(iii) T1 is bijective and T2 is not surjective.(iv) T2 is bijective and T1 is not injective.93.2. OPERATOR COMPOSITIONThen k(T1T2) ⩾ 12 .Proof. Since T1 and T2 are nonexpansive operators, we have T1T2 is non-expansive as well. Each one of the four conditions implies that T1T2 is notbijective. Now use Corollary 2.14.Ogura and Yamada obtained the following result about averagedness ofoperator composition.Fact 3.4. [OY02, Theorem 3](see also [CY15, Proposition 2.4]) Let T1 :X → X be α1-averaged, and let T2 : X → X be α2-averaged, where α1, α2 ∈(0, 1). SetT = T1T2 and α =α1 + α2 − 2α1α21− α1α2 .Then α ∈ (0, 1) and T is α-averaged.Formulating this result here using the modulus of averagedness, we havethe following.Proposition 3.5. Let T1 : X → X and T2 : X → X be nonexpansive.Suppose k (T1) k (T2) ̸= 1. Thenk (T1T2) ⩽k (T1) + k (T2)− 2k (T1) k (T2)1− k (T1) k (T2) .Proof. Let φ (T1, T2) =k(T1)+k(T2)−2k(T1)k(T2)1−k(T1)k(T2) . Consider five cases.(1) Suppose k (Ti) = 1 for some i ∈ {1, 2}. Then φ (T1, T2) = 1. SinceT1 and T2 are nonexpansive, we have T1T2 is nonexpansive, i.e., k (T1T2) ⩽1 = φ (T1, T2).(2) Suppose k (Ti) ∈ (0, 1) for each i ∈ {1, 2}. Then combining Proposi-tion 2.2 and Fact 3.4 we have T1T2 is φ (T1, T2)-averaged. Thus k (T1T2) ⩽φ (T1, T2).(3) Suppose k (T1) = 0 and k (T2) ∈ (0, 1). Then there exists v1 ∈ X suchthat T1 = Id+v1 by Proposition 2.3. Thus T1T2 = T2 + v1 and k (T1T2) =k (T2 + v1) = k (T2) by Proposition 2.1. While φ (T1, T2) = k (T2) in thiscase, we have k (T1T2) = φ (T1, T2).(4) Suppose k (T1) ∈ (0, 1) and k (T2) = 0. Then there exists v2 ∈ Xsuch that T2 = Id+v2 by Proposition 2.3. Thus T1T2 = T1(· + v2) andk (T1T2) = k (T1(·+ v2)) = k (T1) by Proposition 2.1. While φ (T1, T2) =k (T1) in this case, we have k (T1T2) = φ (T1, T2).(5) Suppose k (T1) = k (T2) = 0. Then there exist v1 ∈ X and v2 ∈ Xsuch that T1 = Id+v1 and T2 = Id+v2 by Proposition 2.3. Thus T1T2 =103.2. OPERATOR COMPOSITIONId + v2 + v1 and k (T1T2) = k (Id + v2 + v1) = 0. While φ (T1, T2) = 0 inthis case, we have k (T1T2) = φ (T1, T2).Thus we complete the proof.Proposition 3.6. Let C be a nonempty closed convex set in X and C ̸= X.Then for any nonexpansive operator T : X → X:12⩽ k (T ◦ PC) ⩽ 12− k(T )and12⩽ k (PC ◦ T ) ⩽ 12− k(T ) .Proof. Observe that PC is neither surjective nor injective in this case. Thusby Proposition 3.3 we have k (T ◦ PC) ⩾ 12 and k (PC ◦ T ) ⩾ 12 . Now byTheorem 2.6k (T ) + k (PC)− 2k (T ) k (PC)1− k (T ) k (PC) =12− k(T ) .Thus by Proposition 3.5 we have k (T ◦ PC) ⩽ 12−k(T ) and k (PC ◦ T ) ⩽12−k(T ) , which complete the proof.Remark 3.7. Particularly, if we let T = PV and C = U , where U and Vare both closed linear subspaces, then k (PV PU ) =1+cF2+cF∈ [12 , 23], wherecF ∈ [0, 1] (see [BBM23, Corollary 3.3]). This coincides with the bounds weobtained as 12−k(PU ) =23 combining Proposition 3.6 and Theorem 2.6.We can generalize the results of composition of two operators to compo-sition of finitely many operators.Proposition 3.8. Let m ⩾ 2 be an integer and let I = {1, . . . ,m}. For anyi ∈ I, let Ti be nonexpansive from X to X. Suppose one of the followingholds:(i) T1 is not surjective.(ii) Tm is not injective.(iii) T1 is bijective and T2 · · ·Tm is not surjective.(iv) Tm is bijective and T1 · · ·Tm−1 is not injective.Then k (T1 · · ·Tm) ⩾ 12 .Proof. Directly by Proposition 3.3.Corollary 3.9. Let C1, . . . , Cm be nonempty closed convex sets in X. IfC1 ̸= X or Cm ̸= X, then k (PC1 · · ·PCm) ⩾ 12 .113.3. LIMITING OPERATOR3.3 Limiting operatorRecall that a sequence (xn)n∈N in a Hilbert space X is said to convergeweakly to a point x in X if for each y in Xlimn→∞ ⟨xn, y⟩ = ⟨x, y⟩.We use the notation limwn→∞ xn to be the weak limit of (xn)n∈N.The following fact shows that averaged operators perform very well inalgorithms.Fact 3.10. [BC17, Proposition 5.16] Let α ∈ (0, 1) and let T : X → X beα-averaged such that FixT ̸= ∅. Then for any x ∈ X, (Tnx)n∈N convergesweakly to a point in FixT .Definition 3.11. (limiting operator) Let α ∈ (0, 1) and let T : X → X beα-averaged such that FixT ̸= ∅. Define its limiting operator T∞ : X → XbyT∞x :=wlimn→∞Tnx.Remark 3.12. The full domain and single-valuedness of T∞ in this setting,are guaranteed by Fact 3.10 and the fact that weak limit is unique. Hencewe can write T∞ : X → X.Example 3.13. [BC17, Example 5.29] Let α ∈ (0, 1) and let T : X → Xbe α-averaged such that C := FixT ̸= ∅ and C ̸= X. Suppose T is linear.Then T∞ = PC in which case k (T∞) = 12 .Example 3.14. [BC17, Proposition 5.9] Let α ∈ (0, 1) and let T : X → Xbe α-averaged such that C := FixT ̸= ∅. Suppose C is a closed affinesubspace of X. Then T∞ = PC .Lemma 3.15. [BC17, Corollary 4.24 and Corollary 5.8] Let α ∈ (0, 1) andlet T : X → X be α-averaged such that C := FixT ̸= ∅. Then C is anonempty closed convex set in X. Moreover, for each x ∈ X, PCTnx→ T∞xconverges strongly.Theorem 3.16. Let α ∈ (0, 1) and let T : X → X be α-averaged such thatC := FixT ̸= ∅ and C ̸= X. Thenk (T∞) ∈[12, 1].123.3. LIMITING OPERATORProof. Since T : X → X is nonexpansive, we have(∀x ∈ X)(∀y ∈ X) ∥Tx− Ty∥ ⩽ ∥x− y∥.By induction we have for any n ∈ N,(∀x ∈ X)(∀y ∈ X) ∥Tnx− Tny∥ ⩽ ∥x− y∥.By Lemma 3.15, C is a nonempty closed convex set in X and C ̸= X. ThusPC is nonexpansive and we have for any n ∈ N,(∀x ∈ X)(∀y ∈ X) ∥PCTnx− PCTny∥ ⩽ ∥Tnx− Tny∥ ⩽ ∥x− y∥.Since ∥ · ∥ is continuous, letting n→∞ we have(∀x ∈ X)(∀y ∈ X)∥∥∥ limn→∞PCTnx− limn→∞PCTny∥∥∥ ⩽ ∥x− y∥.Again by Lemma 3.15,(∀x ∈ X)(∀y ∈ X) ∥T∞x− T∞y∥ ⩽ ∥x− y∥.Thus T∞ is nonexpansive, i.e., k (T∞) ∈ [0, 1].On the other hand, ranT∞ = FixT ̸= X by Fact 3.10. Thus applyingCorollary 2.14 we have k(T∞) ⩾ 12 .In the following, we discuss the averagedness of limiting operator on R.In Example 3.13 we see that if T : X → X is α-averaged and linear, thenk (T∞) = 12 . The following example shows that this is not true in nonlinearcase.Example 3.17. Letf(x) :=0 if x ⩽ 0,x if 0 ⩽ x ⩽ 1,−12x+ 32 if x ⩾ 1.Then f is 34 -averaged and FixT = [0, 1]. However,f∞(x) =0 if x ⩽ 0 or x ⩾ 3,x if 0 ⩽ x ⩽ 1,−12x+ 32 if 1 ⩽ x ⩽ 3,and k (f∞) = 34 .133.3. LIMITING OPERATORProof. We can check 43(f − 14 Id)is nonexpansive, which implies f is 34 -averaged. We can also check 43(f∞ − 14 Id)is nonexpansive, which impliesf∞ is 34 -averaged. Now for any ε > 0, defineN :=f∞ −(14 + ε)Id34 − ε.Then one can check that N is not nonexpansive for any ε > 0. This impliesthat f∞ is not (34 − ε)-averaged. Thus k (f∞) = 34 .Next, we show that if T : X → X is firmly nonexpansive, which is astronger condition than averagedness, then on the real line it is true thatk (T∞) = 12 .Proposition 3.18. Let f : R→ R be firmly nonexpansive such that f ̸= Idand Fix f ̸= ∅. Suppose f is differentiable on R. Then f∞ is speciallynonexpansive, i.e.,k (f∞) =12.Proof. Since f is firmly nonexpansive, i.e., k(f) ⩽ 12 , we have infx∈R f ′(x) ∈[0, 1] by Proposition 2.4. Thus for any x ∈ R, f ′(x) ∈ [0, 1], since f is alsononexpansive. Therefore,k(f2)=1− infx∈R(f2)′(x)2=1− infx∈R f ′(f(x)) · f ′(x)2⩽ 12.By induction and the chain rule for the derivative, we can show that for anyn ∈ N,k (fn) ⩽ 12,i.e.,(∀x ∈ R)(∀y ∈ R) ∥fn(x)− fn(y)∥2 ⩽ ⟨x− y, fn(x)− fn(y)⟩.Now let n→∞, then(∀x ∈ R)(∀y ∈ R) ∥f∞(x)− f∞(y)∥2 ⩽ ⟨x− y, f∞(x)− f∞(y)⟩ ,since norm is continuous. Thus k (f∞) ⩽ 12 .The proof completes by applying Theorem 3.16.In fact we can drop the smooth condition in the proposition above.143.3. LIMITING OPERATORProposition 3.19. Let f : R→ R be firmly nonexpansive such that f ̸= Idand Fix f ̸= ∅. Then f∞ = PFix f and k (f∞) = 12 .Proof. Since Fix f ⊆ R is nonempty, closed and convex, it must take theform [a,+∞), (−∞, b] or [a, b] with a, b ∈ R because f ̸= Id. Because theproofs for all cases are similar, let us assume that Fix f = [a, b]. Whenx ⩾ b, because f is nondecreasing (since firmly nonexpansive), we havef(x) ⩾ f(b) = b, f2(x) ⩾ f(b) = b, and an induction leads fn(x) ⩾ b.Then f∞(x) ⩾ b by Fact 3.10. Since f∞(x) ∈ [a, b] by Fact 3.10 again, wederive that f∞(x) = b. Similar arguments give f∞(x) = a when x ⩽ a.Clearly, when x ∈ [a, b], (∀n ∈ N)fn(x) = x, so f∞(x) = x. Altogetherf∞ = PFixf .Due to Example 3.13 and Proposition 3.19, we propose the followingquestion: Let T : X → X be firmly nonexpansive such that T ̸= Id andFixT ̸= ∅. Then do we have T∞ is specially nonexpansive?15Chapter 4ResolventIn this chapter we will use monotone operator theory to analyze themodulus of averagedness. We will give a characterization of normal andspecial nonexpansiveness of resolvents. We will define various values as-sociated with maximally operators and provide a formula to calculate themodulus of resolvents.The main new results of this chapter are Theorem 4.11, Theorem 4.20and Theorem 4.27.Let A : X ⇒ X be a set-valued operator. Recall that the resolvent of A isJA = (Id+A)−1 and the reflected resolvent of A is RA = 2JA−Id. The graphof A is gra A = {(x, u) ∈ X ×X | u ∈ Ax} and the inverse of A, denoted byA−1, is the operator with graph graA−1 = {(u, x) ∈ X ×X | u ∈ Ax}. Thedomain of A is domA = {x ∈ X | Ax ̸= ∅}. We say A is monotone if∀(x, u), (y, v) ∈ graA, ⟨x− y, u− v⟩ ⩾ 0.We say A is maximally monotone, if it is monotone and there is no monotoneoperator B : X ⇒ X such that graA is properly contained in graB.4.1 PreparationFact 4.1. (Minty’s Theorem)[BC17, Proposition 23.8] Let T : X → X.Then T is firmly nonexpansive if and only if T is the resolvent of a maximallymonotone operator.Lemma 4.2. Let T : X → X be nonexpansive. Then for any β ∈ [0, 1], theformula holds:k((1− β)Id + βT ) = βk(T ).Proof. Suppose k(T ) = 0. Then by Proposition 2.3, there exists v ∈ Xsuch that T = Id+v. Thus (1− β) Id+βT = Id+βv, which implies k((1−β) Id+βT ) = 0.Suppose k(T ) > 0. Set N = (1 − (k(T ))−1)Id + (k(T ))−1T . ThenT = (1− k(T )) Id+k(T )N and (1− β) Id+βT = (1− βk(T )) Id+βk(T )N ,which implies k((1− β) Id+βT ) = βk(T ).164.1. PREPARATIONDefinition 4.3. (Yosida regularization)Let A : X ⇒ X be maximallymonotone and let µ > 0. The Yosida µ-regularization of A is the operatorYµ(A) :=(µ Id+A−1)−1.Fact 4.4. [RW04, Lemma 12.14]Let µ > 0. Then the following identityholds:Yµ(A) =1µ(Id−JµA) .The following result is [BC17, Proposition 23.7(iv)]. Here we take theopportunity to give a detailed proof.Proposition 4.5. Let A : X ⇒ X be maximally monotone and let µ, α > 0.ThenJαYµ(A) =µµ+ αId+αµ+ αJ(µ+α)A.Proof. Firstly,αYµ(A) = α(µ Id+A−1)−1=[(µ Id+A−1)(α−1 Id)]−1=(α−1µ Id+A−1(α−1 Id))−1=(α−1µ Id+(αA)−1)−1= Yα−1µ(αA).Thus we only need to prove the formula holds for α = 1.Let y ∈ X, z = J(µ+1)A(y) and x = µµ+1y + 1µ+1z. We will provex = JYµ(A)(y). We have z = (µ + 1)x − µy, y − z = µ+1µ (x − z) andYµ(A) =1µ (Id−JµA). Thusz = J(µ+1)A(y)⇔ y − z ∈ (µ+ 1)Az⇔ µ+ 1µ(x− z) ∈ (µ+ 1)Az⇔ x− z ∈ (µA)z⇔ z = JµAx⇔ (µ+ 1)x− µy = JµAx⇔ y − x = x− JµAxµ= Yµ(A)(x)⇔ x = JYµ(A)(y).174.2. CHARACTERIZATION OF NORMALLY NONEXPANSIVE OPERATORSCombining Proposition 4.5 and Lemma 4.2, we have the following.Corollary 4.6. Let A : X ⇒ X be maximally monotone and α ∈ [0, 1).Then(i) JαY1−α(A) = (1− α)Id + αJA.(ii) k(JαY1−α(A)) = αk(JA).Corollary 4.7. Let A : X ⇒ X be maximally monotone and µ > 0. Then(i) JYµ(A) =µµ+1 Id+1µ+1J(µ+1)A.(ii) k(JYµ(A)) =1µ+1k(J(µ+1)A).k ⩽ 1IdJAJαY1−α(A)4.2 Characterization of normally nonexpansiveoperatorsLet µ > 0. Recall that T : X → X is µ-cocoercive if µT is firmlynonexpansive, i.e.,(∀x ∈ X)(∀y ∈ X) ⟨x− y, Tx− Ty⟩ ⩾ µ∥Tx− Ty∥2.Fact 4.8. [BC17, Proposition 23.21] T : X → X is µ-cocoercive if and onlyif there exists a maximally monotone operator A, such that T = Yµ(A).Lemma 4.9. Let A : X ⇒ X be maximally monotone. Suppose JA isnormally nonexpansive. Then A is single-valued with full domain, and co-coercive.184.2. CHARACTERIZATION OF NORMALLY NONEXPANSIVE OPERATORSProof. Without loss of generality k (JA) ̸= 0. LetT :=JA − (1− 2k (JA)) Id2k (JA).Then we have JA = (1− 2k (JA)) Id+2k (JA)T . By Lemma 4.2, we havek(T ) = 12 . Since T is firmly nonexpansive, by Fact 4.1, there exists amaximally monotone operator B, such that T = JB. Thus by Corollary4.6 we haveJA = (1− 2k (JA)) Id+2k (JA)T= (1− 2k (JA)) Id+2k (JA) JB= J2k(JA)Y1−2k(JA)(B).Therefore, A = 2k (JA)Y1−2k(JA)(B). Since JA is normally nonexpansive,we have 2k (JA) ∈ (0, 1). Thus 2k (JA)Y1−2k(JA)(B) is a single-valued, fulldomain, and cocoercive operator due to Fact 4.8. Hence A is single-valuedwith full domain, and cocoercive.Lemma 4.10. Let A : X ⇒ X be maximally monotone. Suppose A is single-valued with full domain, and cocoercive. Then JA is normally nonexpansive.Proof. Since A is single-valued with full domain, and cocoercive, by Fact4.8, there exist a maximally monotone operator B and µ > 0, such thatA = Yµ(B). Since B is maximally monotone, by Corollary 4.7, we haveJYµ(B) =µµ+ 1Id+1µ+ 1J(µ+1)B.ThusJA =µµ+ 1Id+1µ+ 1J(µ+1)B.Since B is maximally monotone and µ+1 > 1, we have (µ+1)B is maximallymonotone as well. Thus k(J(µ+1)B) ⩽ 12 by Fact 4.1. Now Lemma 4.2 givesk (JA) =1µ+ 1k(J(µ+1)B)⩽ 1µ+ 1· 12<12,i.e., JA is normally nonexpansive.Theorem 4.11. (characterization of normally nonexpansive opera-tors)Let A : X ⇒ X be maximally monotone. Then JA is normally nonex-pansive if and only if A is single-valued with full domain, and cocoercive.Proof. Combine Lemma 4.9 and Lemma 4.10.194.3. COCOERCIVE VALUERemark 4.12. There are some propositions in this section that may be sim-plified compared to [BMW12].Corollary 4.13. Let A : X ⇒ X be maximally monotone. Then JA isspecially nonexpansive if and only if A does not satisfy one of the followingconditions:(i) single-valued;(ii) full domain;(iii) cocoercive.4.3 Cocoercive valueIt can be verified that if an operator is µ0-cocoercive, then it is µ-cocoercive if µ ⩽ µ0. Observing this property, we define the following valuefor a maximally monotone operator.Definition 4.14. (cocoercive value) Let A : X ⇒ X be maximally mono-tone. If A is single-valued with full domain, and cocoercive, then define thecocoercive value of A byc(A) := sup{µ > 0 | A is µ-cocoercive }.Otherwise, we define c(A) = 0.The following properties of cocoercive value can be directly verified,hence we omit the proof.Proposition 4.15. Let A : X ⇒ X be maximally monotone and µ, α > 0.(i) Suppose c(A) ∈ (0,+∞). Then A is c(A)-cocoercive. Moreover, A is1c(A) -Lipschitz on X.(ii) c(A) = +∞ if and only if A is a constant operator.(iii) c(αA) = α−1c(A).Now we turn to Yosida regularization.Lemma 4.16. Let A : X ⇒ X and B : X ⇒ X be maximally monotoneoperators. Let µ > 0 and suppose that Yµ(A) = Yµ(B). Then A = B.Proof. By Fact 4.4,Yµ(A) = Yµ(B)⇔ µ−1 (Id−JµA) = µ−1 (Id−JµB)⇔ JµA = JµB⇔ µA = µB⇔ A = B.204.3. COCOERCIVE VALUERemark 4.17. Combining Fact 4.8 and Lemma 4.16, we can see Yµ is abijective mapping between operator spaces. Thus for a cocoercive operatorA, (Yµ)−1A is well-defined.Lemma 4.18. [BC17, Proposition 23.7] Let A : X ⇒ X be maximallymonotone and µ, γ > 0. ThenYµ(Yγ(A)) = Y(µ+γ)(A).Proof. By definition for any x ∈ X,y = Y(µ+γ)(A)(x)⇔ y =((µ+ γ) Id+A−1)−1(x)⇔ x ∈ (µ+ γ)y +A−1y⇔ y ∈ A(x− (µ+ γ)y)⇔ y ∈ A((x− µy)− γy)⇔ y = Yγ(A)(x− µy)⇔ y = Yµ (Yγ(A)) (x).Lemma 4.19. Let A : X ⇒ X be maximally monotone. Suppose c(A) ∈(0,+∞). Let µ := c(A). Then (Yµ)−1A is not cocoercive.Proof. Since µ = c(A) ∈ (0,+∞), by Proposition 4.15, A is µ-cocoercive.Thus by Remark 4.17 there exists a unique B such that A = Yµ(B). Assumethat B is γ-cocoercive for some γ > 0. Then again by Remark 4.17 thereexists a unique maximally monotone operator C such that B = Yγ(C). Thusby Lemma 4.18A = Yµ(B)= Yµ (Yγ(C))= Y(µ+γ)(C).Therefore by Fact 4.8, we have A is (µ+γ)-cocoercive, which implies µ+γ ⩽c(A). But c(A) = µ, which leads to γ ⩽ 0, contradiction.Now we are able to propose the following formula.Theorem 4.20. (modulus of averagedness via cocoercive value) LetA : X ⇒ X be maximally monotone. Thenk (JA) =1211 + c(A).214.3. COCOERCIVE VALUEProof. Suppose c(A) = +∞. Then there exists a vector v ∈ X such thatA(x) = v for all x ∈ X. ThusJA = (Id+A)−1= (Id+v)−1= Id−vand we have k (JA) = 0. Since1211 + c(A)= 0,the formula holds.Suppose c(A) ∈ (0,+∞). Let µ := c(A). Then by Fact 4.8 and Lemma4.19, B = (Yµ)−1A is maximally monotone but not cocoercive. Since µ+1 >0, we have (µ + 1)B is maximally monotone but not cocoercive. Thus byCorollary 4.13 k(J(µ+1)B)= 12 . On the other hand, by Corollary 4.7, wehaveJYµ(B) =µµ+ 1Id+1µ+ 1J(µ+1)B,i.e.,JA =µµ+ 1Id+1µ+ 1J(µ+1)B.Using formula in Lemma 4.2, the result followsk (JA) =1µ+ 1k(J(µ+1)B)=121µ+ 1=121c(A) + 1.Suppose c(A) = 0. Then A is not cocoercive. By Corollary 4.13 we havek (JA) =12 . Since1211 + c(A)=12because c(A) = 0, the formula still holds.Remark 4.21. Theorem 4.20 will later be generalized in Proposition 6.27.Adding a parameter α > 0, we have the following.224.4. MONOTONE VALUECorollary 4.22. Let A : X ⇒ X be maximally monotone and α > 0. Thenk (JαA) =12αα+ c(A).Proof. Combining Proposition 4.15 and Theorem 4.20 we havek (JαA) =1211 + c(αA)=1211 + α−1c(A)=12αα+ c(A).4.4 Monotone valueLet β > 0. Recall that we say A : X ⇒ X is β-strongly monotone ifA− βId is monotone, i.e.,(∀(x, u) ∈ graA)(∀(y, v) ∈ graA) ⟨x− y, u− v⟩ ⩾ β∥x− y∥2.It can be verified that if an operator is β0-strongly monotone, then itis β-strongly monotone if β ⩽ β0. Observing this property, we define thefollowing function for a maximally monotone operator.Definition 4.23. (monotone value) Let A : X ⇒ X be maximally mono-tone. If A is strongly monotone, then define the monotone value of A bym(A) := sup{β > 0 | A is β-strongly monotone }.Otherwise, we define m(A) = 0.The following properties of monotone value can be directly verified, hencewe omit the proof.Proposition 4.24. Let A : X ⇒ X and B : X ⇒ X be maximally mono-tone operators. Let α, µ ⩾ 0. Then(i) m(αA) = αm(A).(ii) m(A+ µId) = m(A) + µ.(iii) m(A+B) ⩾ m(A) +m(B).234.4. MONOTONE VALUEThe following lemma is known and can be quickly deduced from defini-tions.Lemma 4.25. Let A : X ⇒ X be maximally monotone and β > 0. Then Ais β-cocoercive if and only if A−1 is β-strongly monotone.The following shows the natural duality between cocoercive value andmonotone value.Proposition 4.26. (duality) Let A : X ⇒ X be maximally monotone.Then(i) c (A) = m(A−1).(ii) c(A−1)= m (A).Proof. Since(A−1)−1= A, we only need to prove (1).If A is β-cocoercive for some β > 0, then m(A−1) = c (A) by Lemma4.25.If A is maximally monotone but not β-cocoercive for any β > 0, then A−1is not β-strongly monotone for any β > 0 again by Lemma 4.25. By the def-inition of cocoercive value and monotone value, we have c(A) = m(A−1)=0.The following characterizes when k (JA) = k (JA−1).Theorem 4.27. Let A : X ⇒ X be maximally monotone. Then k (JA) =k (JA−1) if and only if c(A) = m(A).Proof. By Theorem 4.20 and Proposition 4.26:k (JA) = k (JA−1)⇔1211 + c(A)=1211 + c (A−1)⇔ 1211 + c(A)=1211 +m(A)⇔ c(A) = m(A).It is well-known that the Yosida regularization Yµ(A) is µ-cocoercive forµ > 0. The following characterizes how cocoercive Yµ(A) could be.Proposition 4.28. Let A : X ⇒ X be maximally monotone and µ > 0.Then c (Yµ(A)) = c(A) + µ.244.4. MONOTONE VALUEProof. Since Yµ(A) =(µId +A−1)−1, combining Proposition 4.26 and Propo-sition 4.24 (ii) we havec (Yµ(A)) = c((µId +A−1)−1)= m(µId +A−1)= µ+m(A−1)= µ+ c(A).Theorem 4.29. Let A : X ⇒ X be maximally monotone and α > 0. Thenk (JαA) c (Yα(A)) =α2.In particular,k (JA) c (JA−1) =12.Proof. Combining Corollary 4.22 and Proposition 4.28 we obtain the result.Corollary 4.30. Let T : X → X be firmly nonexpansive. Thenk (T ) c (Id−T ) = 12.Proof. By Fact 4.4, JA−1 = Id−JA. The result follows by combining Fact4.1 and Theorem 4.29.25Chapter 5Proximal operatorIn this chapter, we will characterize the normal and special nonexpan-siveness of proximal operators. Moreover, we will give a formula to calculatethe modulus of proximal operators.The main new results of this chapter are Theorem 5.4, Corollary 5.5 andTheorem 5.13.Recall that Γ0(X) is the set of all proper lower semicontinuous convexfunctions from X to (−∞,+∞]. The proximal operator of f ∈ Γ0(X) isdefined byPf (x) = argminw∈X{f(w) +12∥w − x∥2}and the domain of f is dom f = {x ∈ X | f(x) < +∞}. It is well-knownthat Pf = (Id+∂f)−1, where ∂f is the subdifferential of f given by ∂f(x) ={u ∈ X | (∀y ∈ X) f(x) + ⟨y − x, u⟩ ⩽ f(y)} if x ∈ dom f , and ∂f(x) = ∅if x /∈ dom f . See, e.g., [BC17] and [RW04].5.1 Characterization of normal proximaloperatorsThe following concept is useful in convex analysis and optimization, see,e.g., [Bec17].Definition 5.1. (L-smoothness) Let L ∈ [0,+∞). Then f is L-smoothon X if f is Frechet differentiable on X and ∇f is L-Lipschitz, i.e.,(∀x ∈ X)(∀y ∈ X) ∥∇f(x)−∇f(y)∥ ⩽ L∥x− y∥.Fact 5.2. (Moreau)[BC17, Theorem 20.25] Let f ∈ Γ0(X). Then ∂f ismaximally monotone.Fact 5.3. (Baillon-Haddad)[BH77](see also [BC17, Corollary 18.17]) Letf ∈ Γ0(X). Suppose f is Frechet differentiable on X. Then ∇f is µ-cocoercive if and only if ∇f is µ−1-Lipschitz continuous.265.2. LIPSCHITZ VALUEFor further properties of L-smooth functions, see [Bec17].Theorem 5.4. (characterization of normal proximal operators) Letf ∈ Γ0(X). Then Pf is normally nonexpansive if and only if there existsL > 0 such that f is L-smooth on X.Proof. By Fact 5.2, ∂f is maximally monotone since f ∈ Γ0(X). Let A = ∂fin Theorem 4.11 and combine it with Fact 5.3.Corollary 5.5. [SW24, Theorem 5.4]Let f ∈ Γ0(X). Then the followingsare equivalent:(1) k (Pf ) <12 , i.e., Pf is normally nonexpansive.(2) f is Frechet differentiable on X and ∇f is Lipschitz continuous (i.e.,Lipschitz smooth).(3) Pf is a bi-Lipschitz self-homeomorphism of X.Corollary 5.6. Let f ∈ Γ0(X). Pf is specially nonexpansive if and only iff is not L-smooth on X for any L > 0.Corollary 5.7. Let f ∈ Γ0(X). Suppose dom f ̸= X. Then Pf is speciallynonexpansive.Proof. dom f ̸= X implies dom ∂f ̸= X. Thus f is not L-smooth for anyL > 0 and the result follows by Corollary 5.6.Remark 5.8. Note that PC = PιC when C is a nonempty closed convexsubset of X, whereιC(x) ={0 x ∈ C,+∞ x /∈ C.see [BM23, Example 25.2]. We have ιC ∈ Γ0(X) and obviously dom ιC = C.Thus by Corollary 5.7, PC is specially nonexpansive when C ̸= X, whichrecovers Theorem 2.6.5.2 Lipschitz valueDefinition 5.9. (Lipschitz value) Let A : X ⇒ X be maximally mono-tone. If A is single-valued with full domain and Lipschitz continuous on X,then define the Lipschitz value of A byℓ(A) := inf {L ⩾ 0 | ∀x, y ∈ X, ∥A(x)−A(y)∥ ⩽ L∥x− y∥} .Otherwise, we define ℓ(A) = +∞.275.2. LIPSCHITZ VALUENote that T : X → X is nonexpansive if and only if ℓ(T ) ⩽ 1. And T isa Banach contraction if and only if ℓ(T ) < 1.The following shows the relationship between Lipschitz value and co-coercive value. Note that we follow the convention that (+∞)−1 = 0 and0−1 = +∞.Lemma 5.10. Let A : X ⇒ X be maximally monotone. Thenℓ(A) ⩽ [c(A)]−1.Proof. Suppose c(A) ∈ (0,+∞). Then by Proposition 4.15, A is [c(A)]−1-Lipschitz on X. Thus ℓ(A) ⩽ [c(A)]−1.Suppose c(A) = +∞. By Proposition 4.15 A is a constant operator.Thus ℓ(A) = 0 = [c(A)]−1.Suppose c(A) = 0. Then ℓ(A) ⩽ +∞ = [c(A)]−1.Fact 5.11. [BC17, Proposition 17.31] Let f be convex and proper on X,and suppose that x ∈ int dom f . Thenf is Gaˆteaux differentiable at x⇔ ∂f(x) is a singletonin which case ∂f(x) = {∇f(x)}.Proposition 5.12. Let f ∈ Γ0(X). Then ℓ(∂f) = [c(∂f)]−1.Proof. Suppose c(∂f) ∈ (0,+∞). Then ∂f is single-valued with full domain.Thus ∂f = ∇f by Fact 5.11. Since ∇f is c(∂f)-cocoercive, by applying Fact5.3 we have ℓ(∇f) = [c(∇f)]−1.Suppose c(∂f) = +∞. Then ∂f is a constant operator by Proposition4.15. Thus ℓ(∂f) = 0 = [c(A)]−1.Suppose c(∂f) = 0. If ∂f is single-valued with full domain, then againby applying Fact 5.11 and Fact 5.3 we have ∂f = ∇f is not Lipschitz,thus ℓ(∂f) = +∞ = [c(∂f)]−1. If ∂f is not single-valued, or not withfull domain, then ℓ(∂f) = +∞ by the definition of Lipschitz value. Thusℓ(∂f) = +∞ = [c(∂f)]−1.Now we are able to propose the following formula.Theorem 5.13. (modulus of averagedness via Lipschitz value) Letf ∈ Γ0(X). Thenk (Pf ) =1211 + [ℓ(∂f)]−1.285.2. LIPSCHITZ VALUEProof. By Fact 5.2, ∂f is maximally monotone. The result follows by lettingA = ∂f in Theorem 4.20 and combining it with Proposition 5.12.Adding a parameter α > 0, we have the following. Note ℓ(αT ) = αℓ(T ).Corollary 5.14. Let f ∈ Γ0(X) and α > 0. Suppose f is Frechet differen-tiable on X and ∇f is Lipschitz. Thenk (Pαf ) =12αℓ(∇f)1 + αℓ(∇f) .In particular,k (Pf ) =12ℓ(∇f)1 + ℓ(∇f) .Example 5.15. Let C be a nonempty closed convex set in X and C ̸= X.Then for any α > 0:(i) k(Pα2d2C)= 12α1+α .(ii) c (Id−PC) = ℓ (Id−PC) = 1.(iii) Pα2d2Cis bijective.Proof. (i) By [Bec17, Example 6.65], Pα2d2C= 11+α Id+α1+αPC . Thus wehave k(Pα2d2C)= α1+αk(PC) =12α1+α by Lemma 4.2 and Theorem 2.6.(ii) Since 12d2C ∈ Γ0(X) and ∇12d2C = (Id−PC), by (i), Corollary 5.14,and Proposition 5.12, c (Id− PC) = ℓ (Id− PC) = 1.(iii) By (i), k(Pα2d2C)= 12α1+α <12 , thus Pα2 d2Cis normally nonexpansive.By Theorem 2.13, Pα2d2Cis bijective.We attach Appendices with three alternative methods to prove Pα2d2Cisbijective.29Chapter 6Conical averagedness andgeneralized monotonicityIn this chapter we will survey conically averaged operators, a naturalgeneralization of averaged operators. We will see most of our previous re-sults can be put in this broader framework while some new results can beobtained. Generalized monotonicity and comonotonicity, as a generalizationof monotonicity and cocoercity, are also examined both in an independentway and in a way related to conical averagedness.The main new results of this chapter are Corollary 6.18, Theorem 6.31and Corollary 6.35.6.1 IntroductionDefinition 6.1. [BMW21, Definition 3.1](see also [BDP22, Definition 2.1])Let T : X → X. We say T is conically k-averaged, if T can be representedas T = (1− k) Id+kN , where N is nonexpansive and k ∈ [0,+∞).Note conical averagedness is also refered as conical nonexpansiveness.Similar to averagedness, when k ∈ (0,+∞), various characterizations ofconically k-averagedness are available including (see [BDP22, Proposition2.2])(∀x ∈ X)(∀y ∈ X) ∥Tx−Ty∥2 ⩽ ∥x−y∥2− 1− kk∥(Id−T )x− (Id−T )y∥2(6.1)and(∀x ∈ X)(∀y ∈ X) ∥Tx−Ty∥2+(1−2k)∥x−y∥2 ⩽ 2(1−k)⟨x−y, Tx−Ty⟩.(6.2)Using the similar argument, we can naturally extend the BBM modulusof averagedness to the following.Definition 6.2. (Bauschke-Bendit-Moursi modulus of conical aver-agedness) Let T : X → X. Then the Bauschke-Bendit-Moursi modulus of306.2. PROPERTIES OF BBM MODULUSconical averagedness is defined byk(T ) := inf{k ∈ [0,+∞) | T is conically k-averaged }.For conciseness, we call it the BBM modulus.Note if T is not conically k-averaged for any k ∈ [0,+∞), then k(T ) =+∞ since inf ∅ = +∞. We say T : X → X is conically averaged, if it isconically k-averaged for some k ∈ [0,+∞).Example 6.3. Let α ∈ R. Thenk(αId) ={1−α2 if α ⩽ 1,+∞ if α > 1.6.2 Properties of BBM modulusSince the BBM modulus of conical averagedness is a natural extension ofthe BBM modulus of averagedness, most results we proved in the previouschapter (such as Bijective Theorem) hold in terms of conical averagedness.Next, we will provide some propositions that will be frequently used, andpropositions that might have some slight differences from previous ones.We will omit most proofs since the proof tricks are almost the same fromprevious ones.Proposition 6.4. Let T : X → X. Thenk(T ) = 0⇔ ∃v ∈ X : T = Id + v.Proposition 6.5. Let T : X → X be conically averaged. If k(T ) > 0, thenT is conically k(T )-averaged.Fact 6.6. [BDP22, Proposition 2.4] Let I be a finite index set. For each i ∈ Ilet Ti : X → X be conically θi-averaged. Let {wi}i∈I ⊆ R++,∑i∈I ωi = 1.Then∑i∈I ωiTi is conically θ-averaged where θ :=∑i∈I ωiθi.Fact 6.7. [BDP22, Proposition 2.5] Let T1 : X → X and T2 : X → X beconically θ1-averaged and conically θ2-averaged, respectively. Suppose thateither θ1 = θ2 = 1 or θ1θ2 < 1. Let ω ∈ R\{0} and setT :=(1ωT2)(ωT1) and θ :={1 if θ1 = θ2 = 1,θ1+θ2−2θ1θ21−θ1θ2 if θ1θ2 < 1.Then T is conically θ-averaged. Moreover,(i) θ = 1 if and only if θ1 = 1 or θ2 = 1.(ii) θ < 1 if and only if θ1 < 1 and θ2 < 1.316.2. PROPERTIES OF BBM MODULUSProposition 6.8. (generalized Ogura-Yamada inequality) Let Ti :X → X be conically averaged for i = 1, 2 and λ ∈ [0, 1]. Then(1) k ((1− λ)T1 + λT2) ⩽ (1− λ)k (T1) + λk (T2).(2) Suppose that k (T1) k (T2) < 1. Thenk (T1T2) ⩽k (T1) + k (T2)− 2k (T1) k (T2)1− k (T1) k (T2) .Proof. (1) Apply Fact 6.6.(2) Combine Fact 6.7 and Proposition 6.4. The proof argument is similarto Proposition 3.5.Proposition 6.9. Let T : X → X be conically averaged. Then for anyβ ⩾ 0k((1− β) Id+βT ) = βk(T ).Proof. The proof argument is similar to Lemma 4.2.Corollary 6.10. Let T : X → X and suppose Id−T is conically averaged.Then for any β ⩾ 0k(Id−βT ) = βk(Id−T ).Proof. The result follows by observing the identity Id − βT = (1 − β)Id +β(Id−T ).Example 6.11. Let C be a nonempty closed convex set in X and let α ⩾ 0.Suppose C is not a singleton. Thenk (Id−αPC) = α2.Proof. Let x, y ∈ C with x ̸= y since C is not a singleton. Then (Id−PC)(x) =(Id−PC)(y) = 0, thus Id−PC is not injective. Applying Lemma 2.5 we havek (Id− PC) ⩾ 12 .On the other hand, since PC is firmly nonexpansive, we have Id − PCis also firmly nonexpansive. Thus k (Id−PC) = 12 . Now apply Corollary6.10.Next we provide a useful formula to calculate BBM modulus via thecocoercive value of its displacement mapping. Note the earliest version ofthe following fact may be found in [Xu11, Proposition 3.4].326.2. PROPERTIES OF BBM MODULUSFact 6.12. [BMW21, Corollary 3.5] Let T : X → X, and let α ∈ (0,+∞).ThenT is α-conically nonexpansive ⇔ Id− T is 12α-cocoercive.Proposition 6.13. Let T : X → X be conically averaged. Thenk(T ) =121c (Id− T ) .Proof. Combine Fact 6.12 and Proposition 6.4.Example 6.14. Let C be a nonempty closed convex set in X and let α > 0.Suppose C is not a singleton. Then c (αPC) = α−1.Proof. Combine Example 6.11 and Proposition 6.13.Corollary 6.15. Let T : X → X be conically averaged. Thenk(T ) ⩾ 12ℓ(Id− T ).Proof. Combining Proposition 6.13 with Lemma 5.10 we havek(T ) =121c(Id− T ) ⩾12ℓ(Id−T ).Corollary 6.16. Let T : X → X be normally nonexpansive. Then Id − Tis a Banach contraction with constant 2k(T ).Proposition 6.17. Let f ∈ Γ0(X). Suppose f is L-smooth on X for someL > 0. Then the displacement operator Id−∇f is conically averaged andk(Id−∇f) = 12ℓ(∇f).Proof. We have ∇f is L−1-cocoercive by Baillon-Haddad Theorem (Fact5.3). Hence Id−∇f is conically averaged by Fact 6.12.Applying Proposition 5.12 we have 1c(∇f) = ℓ(∇f). Thus by Proposition6.13 we have k(Id−∇f) = 12 1c(∇f) = 12ℓ(∇f).336.3. MONOTONE VALUE AND COMONOTONE VALUERecall that for f ∈ Γ0(X), the Fenchel conjugate of f is defined byf∗(y) := supx∈X(⟨x, y⟩ − f(x)) for y ∈ X. We have f∗ ∈ Γ0(X) and theidentity Pf∗ = Id−Pf . Also recall that for f ∈ Γ0(X), the Moreau envelopeis defined by env f(y) := minx∈X(f(x) + 12∥x− y∥2)for y ∈ X. We haveenv f ∈ Γ0(X) and env f is 1-smooth with ∇ env f = Id − Pf = Pf∗ . Onehas the classic Fenchel–Moreau theorem: (f∗)∗ = f for f ∈ Γ0(X). See,e.g., [BC17] and [RW04].We have the following beautiful formulae.Corollary 6.18. (dual formulae) Let f ∈ Γ0(X). Thenℓ (Pf∗) = 2k (Pf ) and ℓ (Pf ) = 2k (Pf∗) .In particular, Pf is normally nonexpansive if and only if Pf∗ is a Banachcontraction.Proof. Since f ∈ Γ0(X), we have env f ∈ Γ0(X) and is 1-smooth. Thusk(Id − ∇ env f) = 12ℓ(∇ env f) by Proposition 6.17. Since ∇ env f = Id −Pf = Pf∗ , we have ℓ (Pf∗) = 2k (Pf ). We complete the proof by applyingFenchel-Moreau theorem.6.3 Monotone value and comonotone valueFrom now on, we will assume A : X ⇒ X is a set-valued operator on Xwith domA ̸= ∅. Firstly let us recall the following definitions.Definition 6.19. [BMW21, Definition 2.4] Let A : X ⇒ X and let ρ ∈ R.We define the following notions:(i) A is ρ-monotone if (∀(x, u) ∈ graA)(∀(y, v) ∈ graA) we have⟨x− y, u− v⟩ ≥ ρ∥x− y∥2.(ii) A is maximally ρ-monotone if A is ρ-monotone and there is no ρ-monotone operator B : X ⇒ X such that gra B properly contains gra A,i.e., for every (x, u) ∈ X ×X,(x, u) ∈ graA⇔ (∀(y, v) ∈ graA)⟨x− y, u− v⟩ ≥ ρ∥x− y∥2.(iii) A is ρ-comonotone if (∀(x, u) ∈ graA)(∀(y, v) ∈ gra A) we have⟨x− y, u− v⟩ ≥ ρ∥u− v∥2.346.3. MONOTONE VALUE AND COMONOTONE VALUE(iv) A is maximally ρ-comonotone if A is ρ-comonotone and there is noρ-comonotone operator B : X ⇒ X such that gra B properly contains graA, i.e., for every (x, u) ∈ X ×X,(x, u) ∈ graA⇔ (∀(y, v) ∈ graA)⟨x− y, u− v⟩ ≥ ρ∥u− v∥2.We point out that when ρ > 0, then a ρ-comonotone operator A isat most single-valued, and the ρ-comonotonicity reduces to ρ-cocoercity ona subset of X. If the ρ-comonotonicity is maximal, then it reduces to ρ-cocoercity on X and we can write A : X → X. In fact, we have thefollowing.Fact 6.20. [RY22, Theorem 15] Let A : X ⇒ X be ρ-comonotone for ρ > 0.Then A is maximally ρ-comonotone if and only if domA = X.We can observe that if A is ρ0-(co)monotone with ρ0 ∈ R, then it is ρ-(co)monotone for any ρ ⩽ ρ0 by the definition. Thus we define the followingvalues for a set-valued operator, which naturally extend the monotone valueand cocoercive value defined in the previous chapter.Definition 6.21. (monotone value and comonotone value) Let A :X ⇒ X. Then the monotone value of A is defined bym(A) := sup{ρ ∈ R | A is ρ-monotone }.The comonotone value of A is defined byc(A) := sup{ρ ∈ R | A is ρ-comonotone }.When A is maximally monotone, the comonotone value coincides withcocoercive value in view of Fact 6.20, and the monotone value coincides withmonotone value in Definition 4.23. Thus we retain the notation.Note for a ρ-monotone operator A, it is m(A)-monotone. Similar for a ρ-comonotone operator. Moreover, the range of the above values is R∪{±∞}since sup∅ = −∞.The following properties of monotone value and comonotone value canbe verified directly, hence we omit the proof.Proposition 6.22. Let A,B : X ⇒ X with m(A),m(B), c(A), c(B) > −∞.Let α > 0 and µ ∈ R. Then(1) (duality) c(A) = m(A−1)and c(A−1)= m(A).(2) A is monotone iff m(A) ⩾ 0 iff c(A) ⩾ 0.(3) m(αA) = αm(A) and c(αA) = α−1c(A).356.3. MONOTONE VALUE AND COMONOTONE VALUE(4) m(αA+ µId) = αm(A) + µ andc (Yµ(A)) = c(A) + µ.(5) m(A+B) ⩾ m(A) +m(B) andc((A−1 +B−1)−1) ⩾ c(A) + c(B).Example 6.23. (1) Let α ∈ R. Then m (α Id) = α andc(α Id) ={α−1 if α ̸= 0,+∞ if α = 0.(2) Let f(x) = ex on R. Then m(f) = c(f) = 0.(3) Let A : X ⇒ X. Then c(A) = +∞ if and only if A = v for somev ∈ X.The following facts generalize Minty’s Theorem.Fact 6.24. [BMW21, Theorem 2.17] Let A : X ⇒ X be ρ-comonotone forsome ρ > −1. Then A is maximally ρ-comonotone ⇔ ran(Id+A) = X.Fact 6.25. [BMW21, Corollary 2.14](see also [DP19, Proposition 3.4])LetA : X ⇒ X be maximally ρ-comonotone for some ρ > −1. Then JA andJA−1 are single-valued and with full domain.Therefore, for a maximally ρ-comonotone operator A with ρ > −1, wecan write JA : X → X. From now on, we will often consider A be amaximally ρ-comonotone operator with ρ > −1 instead of a non-maximalone, since there always exists a maximal extension by Zorn’s Lemma (thisis also the reason why we consider T : X → X to be full domain in previouschapters). One could obviously generalize this to allow for non-maximaloperators.Next, we provide a characterization of conically averaged operators usingcomonotone value.Fact 6.26. [BMW21, Corollary 3.8] Let T : X → X and α ∈ (0,+∞).Then T is α-conically nonexpansive if and only if it is the resolvent of amaximally ρ-comonotone operator A : X ⇒ X, where ρ = 12α − 1 > −1 (i.e., α = 12(ρ+1)).366.3. MONOTONE VALUE AND COMONOTONE VALUEProposition 6.27. Let A : X ⇒ X be maximally ρ-comonotone with ρ >−1. Thenk (JA) =1211 + c(A).In other words, let T : X → X be conically averaged. Thenk (T ) =1211 + c(T−1 − Id) .Proof. Apply Fact 6.26.Proposition 6.28. Let A : X ⇒ X be maximally ρ-comonotone with ρ >−1. Thenk (JA) c (JA−1) =12.Proof. Combine Proposition 6.13, Fact 6.26, and the identity JA−1 = Id−JA.An alternative proof is to combine c (Yµ(A)) = c(A) + µ (let µ = 1 in thiscase and note Y1(A) = JA−1) in Proposition 6.22 and Proposition 6.27.Corollary 6.29. Let A : X ⇒ X be maximally ρ-comonotone with ρ > −1.Then(1) k (JA) = k (JA−1) if and only if c(A) = m(A).(2) k (JJA) =1212+m(A) .Proof. (1) Apply Proposition 6.27 and c(A−1)= m(A) in Proposition 6.22.(2) We have c (JA) = m(Id+A) = 1 + m(A). Now apply Proposition6.27.Next, we combine the generalized Ogura-Yamada inequality with theformula in Proposition 6.27 to obtain the following. Recall that for set-valued operators A,B : X ⇒ X, the composition BA is given by (BA)x :=⋃y∈AxBy for x ∈ X.Proposition 6.30. Let A,B : X ⇒ X be maximally ρ-comonotone withρ > −1. Suppose (1 + c(A))(1 + c(B)) > 14 . Thenc(A+B(Id+A)) ⩾ 4c(A)c(B)− 14(c(A) + c(B) + 1).In particular, the formula holds if c(A), c(B) > −12 .376.3. MONOTONE VALUE AND COMONOTONE VALUEProof. Since A,B : X ⇒ X are maximally ρ-comonotone with ρ > −1, wehave k (JA) =1211+c(A) and k (JB) =1211+c(B) by Proposition 6.27. Thusk (JA) k (JB) < 1 because (1 + c(A))(1 + c(B)) >14 . Thus by Proposition6.8 we havek (JAJB) ⩽k (JA) + k (JB)− 2k (JA) k (JB)1− k (JA) k (JB) .Note (JAJB)−1 − Id = A + B(Id+A). Now plugging k (JA) = 12 11+c(A) ,k (JB) =1211+c(B) and k (JAJB) =1211+c((JAJB)−1−Id) into the above inequal-ity we havec(A+B(Id+A)) ⩾ 4c(A)c(B)− 14(c(A) + c(B) + 1).In the following we prove an inequality about comonotone value, to es-timate how cocoercive is Id+A in terms of cocoercity of A. This result willlater be generalized.Theorem 6.31. (cocoercive inequality) Let A : X ⇒ X be maximallymonotone. Thenc(Id+A) ⩾ c(A)1 + c(A).Proof. For any α > 0, let A′ = α1+αA and B = α Id. Then c (A′) = 1+αα c(A)andc(A′ +B(Id+A′))= c((1 + α)A′ + α Id)= α−1c(Id+1 + ααA′)= α−1c(Id+A).On the other hand, since c (A′) = 1+αα c(A) ⩾ 0 and c(B) = α−1 ⩾ 0, wehavec(A′ +B(Id+A′))⩾ 4α−1c (A′)− 14 (c (A′) + α−1 + 1)by Proposition 6.30. Thusc(Id+A) ⩾ 4c (A′)− α4 (c (A′) + α−1 + 1)=4(1+α)α c(A)− α4(1+αα c(A) + α−1 + 1)=−α2 + 4(1 + α)c(A)4(1 + α)(1 + c(A)).386.3. MONOTONE VALUE AND COMONOTONE VALUELetφ(α) :=−α2 + 4(1 + α)c(A)4(1 + α)(1 + c(A)).Thenφ′(α) = − 14(1 + c(A))· α(α+ 2)(α+ 1)2< 0for c(A) ⩾ 0 and α > 0. Thussupα>0φ(α) = limα→0φ(α) =c(A)1 + c(A).Since c(Id+A) ⩾ φ(α) for any α > 0, we havec(Id+A) ⩾ supα>0φ(α) =c(A)1 + c(A).Example 6.32. Consider the matrixA =(1 00 2).Then c(A) = 12 ⩾ 0,c(A)1+c(A) =13 and c(Id+A) =13 . Thus in this casec(Id +A) = c(A)1+c(A) .Corollary 6.33. Let A : X ⇒ X be maximally monotone. Then(1) m(JA) ⩾ c(A)1+c(A) .(2) for any α, β > 0 we have c(α Id+βA) ⩾ c(A)β+αc(A) .Proof. Combine Proposition 6.22(1)(3) and Theorem 6.31.Corollary 6.34. Let T : X → X be µ-cocoercive for µ > 0. Then Id+T isµ1+µ -cocoercive.The next corollary concerns the relationship between the conical aver-agedness of −T and cocoercity of T .Corollary 6.35. Let T : X → X be µ-cocoercive for µ > 0. Then −T isconically averaged andk(−T ) ⩽ 12c(T )+12.Similarly, if −T is cocoercive, thenk(T ) ⩽ 12c(−T ) +12.396.3. MONOTONE VALUE AND COMONOTONE VALUEProof. By Theorem 6.31 c(Id+T ) ⩾ c(T )1+c(T ) > 0, thus by Proposition 6.13k(−T ) = 12c(Id+T )⩽ 12 c(T )1+c(T ))=12c(T )+12.Therefore if T is µ-cocoercive, then −T is ( 12µ + 12)-conically averaged 3.3This is also known as(12µ+ 12)-negatively averaged (see [Gis17, Definition 3.7]).40Chapter 7Linear worldIn this chapter we will discuss conically averaged operators, generalizedmonotonicity and comonotonicity in a linear setting. As an application, wewill also discuss the angle between subspaces. Various new formulae areobtained.The main new results of this chapter are Theorem 7.3, Theorem 7.7,Theorem 7.12, Theorem 7.14, Theorem 7.16 and Theorem 7.20.For simplicity, in this chapter we will work in Rn, while most resultscan be generalized to a general Hilbert space setting. Let Mn(R) denotethe set of all n × n matrices, Sn := {A ∈Mn(R) | A⊤ = A} be the set ofall symmetric matrices, and λ(A) denote the set of all eigenvalues of A.Let AS :=A+A⊤2 be the symmetric part of A and λmin(A) be the smallesteigenvalue of A provided all eigenvalues of A are real.7.1 Linear worldFact 7.1. [BMW21, Proposition 5.1] Let A ∈ Rn×n and let ρ ∈ R. Thenthe following hold:(i) A is ρ-monotone ⇔ λmin (AS) ≥ ρ.(ii) A is ρ-comonotone ⇔ λmin(AS − ρA⊤A) ≥ 0.Proposition 7.2. Let A ∈Mn(R). Then(1) m(A) = λmin(AS).(2) c(A) = λmin((A−1)S) if A is invertible.Proof. (1) Apply Fact 7.1.(2) Apply c(A) = m(A−1)in Proposition 6.22.Next we provide a method to compute BBM modulus of a square matrixunder some nonsingular condition.Theorem 7.3. Let A ∈ Mn(R) and suppose that Id−A is invertible (i.e.,1 /∈ λ(A) ). Then A is conically averaged if and only if ((Id−A)−1)Sis417.2. SINGULAR CASEpositive definite, in which case the formula holdsk(A) =121λmin (((Id−A)−1)S).Proof. Combine k(T ) = 12c(Id−T ) in Proposition 6.13 and Proposition 7.2.Example 7.4. Suppose that X = R2 and consider the matrixM :=12[1 01 0].ThenId−M =[12 0−12 1],which is invertible and(Id−M)−1 =[2 01 1].Thus ((Id−M)−1)S=[2 1212 1]andλmin(((Id−M)−1)S)=3−√22.Thereforek(M) =121λmin (((Id−M)−1)S)=3 +√27.This coincides with [BBM23, Example 3.5].7.2 Singular caseIn this section we focus on how to compute BBM modulus of a squarematrix A when 1 ∈ λ(A). A possible approach might be choosing a se-quence of matrices An with 1 /∈ λ(An) that approximate A to calculatek(A). Nonetheless, the following example shows that this approach may failwhen the sequence is not chosen appropriately.Example 7.5. k(Id) ̸= limε→0+ k(Id+ε Id).427.2. SINGULAR CASEProof. This is becausek(αId) ={1−α2 if α ⩽ 1,+∞ if α > 1.However, the following lemma provides a class of matrices that can beused.Lemma 7.6. Let A ∈Mn(R) be conically averaged and suppose that Id−Ais not invertible (i.e., 1 ∈ λ(A)). Then limε→0+ k(A− ε Id) exists andk(A) = limε→0+k(A− ε Id).Proof. Let α := k(A) ∈ [0,+∞). Without loss of generality we may assumeα > 0, thus A is always α-conically averaged. So there exists a nonexpansiveoperator N such that A = (1− α) Id+αN . Thus for any ε > 0, we haveA− ε Id = (1− α) Id+αN − ε Id= (1− α− ε) Id+αN= (1− (α+ ε)) Id+αN= (1− (α+ ε)) Id+(α+ ε) ·(αα+ εN)which implies A−εId is (α+ε)-conically averaged. Thus k(A−ε Id) ⩽ α+ε.Let ε→ 0, we havelimε→0+k(A− ε Id) ⩽ α.Again without loss of generality we may assume k(A−εId) > 0, then A−εIdis k(A− εId)-conically averaged. Therefore for any z ∈ Rn∥(A−ε Id)z∥2+(1− 2k(A− ε Id))∥z∥2 ⩽ 2(1− k(A− ε Id))⟨z, (A− ε Id)z⟩ .Let ε→ 0, then due to continuity of norm and inner product, we have∥Az∥2 + (1− 2 limε→0+k(A− ε Id))∥z∥2 ⩽ 2(1− limε→0+k(A− ε Id))⟨z,Az⟩which implies that A is limε→0+ k(A− εId)-conically averaged. Thusk(A) ⩽ limε→0+k(A− εId).437.2. SINGULAR CASEThereforek(A) ⩽ limε→0+k(A− εId) ⩽ limε→0+k(A− εId) ⩽ k(A)and all inequalities turn to equalities.Now we are able to deal with the singular case.Theorem 7.7. (singular case) Let A ∈ Mn(R) and suppose that Id − Ais not invertible (i.e., 1 ∈ λ(A) ). Then A is conically averaged if and onlyif limε→0+ λmin((((1 + ε) Id−A)−1)S)> 0 in which case the formula holdsk(A) =121limε→0+ λmin ((((1 + ε)Id−A)−1)S).Proof. Since |λ(A)| <∞, there exists γ > 0 such that (1, 1+γ)∩λ(A) = ∅.Thus for any 0 < ε < γ, Id−(A − ε Id) is invertible. Now apply Theorem7.3 and Lemma 7.6.Example 7.8. LetM := 1 0 00 12 00 12 0 .Then M /∈ Sn, 1 ∈ λ(M) and Id−M is not invertible. We have for anyε > 0((1 + ε) Id−M)−1 = ε 0 00 12 + ε 00 −12 1 + ε−1= 1ε 0 00 22ε+1 00 12ε2+3ε+11ε+1thus (((1 + ε) Id−M)−1)S= 1ε 0 00 22ε+1 14ε2+6ε+20 14ε2+6ε+21ε+1andλ((((1 + ε)Id−M)−1)S)={1ε,4ε−√2 + 34ε2 + 6ε+ 2,4ε+√2 + 34ε2 + 6ε+ 2}.447.3. ANGLE BETWEEN SUBSPACESThereforeλmin((((1 + ε)Id−M)−1)S)=4ε−√2 + 34ε2 + 6ε+ 2.Taking the limit we havelimε→0+λmin((((1 + ε)Id−M)−1)S)= limε→0+4ε−√2 + 34ε2 + 6ε+ 2=3−√22> 0.Thus by Theorem 7.7 M is conically averaged andk(M) =12 limε→0+ λmin ((((1 + ε)Id−M)−1)S)=13−√2 =3 +√27.7.3 Angle between subspacesRecall that the cosine of the Dixmier angle [Dix49] (also known as min-imal angle) of two closed linear subspaces U, V of a Hilbert space X iscD(U, V ) := sup{⟨u, v⟩ | u ∈ U, v ∈ V, ∥u∥ ≤ 1, ∥v∥ ≤ 1}and the cosine of the Friedrichs angle [Fri37] iscF (U, V ) := cD(U ∩ (U ∩ V )⊥, V ∩ (U ∩ V )⊥).Note cD(U, V ) = cF (U, V ) if U ∩ V = 0. The cosine of these angles hasa significant impact on describing convergence rates for projection methodssuch as the cyclic projection algorithm. For more on this topic, see [DH06],[Deu95] and [Deu01].The following fact connects BBM modulus with Friedrichs angle.Fact 7.9. [BBM23, Corollary 3.3] Let U, V be closed linear subspaces of X.Then either U = V = X and k (PV PU ) = 0, ork (PV PU ) =1 + cF (U, V )2 + cF (U, V ).Fact 7.10. [BC17, Corollary 4.51] Let T1, . . . , Tn be averaged operators onX such that C :=⋂ni=1 Fix Ti ̸= ∅. Then Fix (Tn · · ·T2T1) = C.Lemma 7.11. Let U, V be closed linear subspaces of X. Let x ∈ X. ThenPV PUx = x if and only if x ∈ U ∩ V .457.3. ANGLE BETWEEN SUBSPACESProof. Since PV and PU are averaged nonexpansive (even firmly nonex-pansive), and (FixPV ) ∩ (FixPU ) = U ∩ V ̸= ∅, we have FixPV PU =(FixPV ) ∩ (FixPU ) = U ∩ V by Fact 7.10.When U∩V ̸= 0, Dixmier angle is known to be trivial as cD(U, V ) = 1 byapplying Cauchy-Schwarz inequality. Next we will propose a formula aboutDixmier angle when U ∩ V = 0, which connects this optimization problemwith eigenvalue.Theorem 7.12. (Dixmier angle formula) Let U, V be linear subspacesof Rn.(1) If U ∩ V ̸= 0, then cD(U, V ) = 1.(2) If U ∩ V = 0, thencD(U, V ) =12λmin(((Id− PV PU )−1)S)− 1− 1.Proof. If U ∩V ̸= 0, cD(U, V ) = 1 follows by taking the unit vector in U ∩Vand applying Cauchy-Schwarz inequality.If U ∩ V = 0, we have 1 /∈ λ(PV PU ) by Lemma 7.11. Thus by Theorem7.3k(PV PU ) =121λmin (((Id− PV PU )−1)S).Also we havek (PV PU ) =1 + cF (U, V )2 + cF (U, V )by Fact 7.9. Since cD(U, V ) = cF (U, V ) when U ∩ V = 0, the result followsby combining these equalities.The following example illustrates Theorem 7.12.Example 7.13. In R2, let V be x-axis and U be a line with slope tan θ,θ ∈ (0, π2 ). ThenPV =(1 00 0)andPU =((cos θ)2 cos θ sin θsin θ cos θ (sin θ)2).We havePV PU =((cos θ)2 cos θ sin θ0 0)467.3. ANGLE BETWEEN SUBSPACESandId−PV PU =((sin θ)2 − cos θ sin θ0 1).Thus(Id− PV PU )−1 =(1(sin θ)2cos θsin θ0 1)and((Id− PV PU )−1)S =(1(sin θ)2cos θ2 sin θcos θ2 sin θ 1).Thereforeλmin(((Id− PV PU )−1)S)=1 + (sin θ)2 − cos θ2(sin θ)2andcD(U, V ) =12λmin((((Id−PV PU )−1)S)− 1− 1=(sin θ)21 + (sin θ)2 − cos θ − (sin θ)2 − 1= cos θwhich implies that the Dixmier angle between U and V is θ.xyV : x-axisU : y = tan θ · xθNext, we focus on Friedrichs angle.477.3. ANGLE BETWEEN SUBSPACESTheorem 7.14. (Friedrichs angle formula) Let U, V be linear subspacesof Rn.(1) If U ∩ V = Rn, then cF (U, V ) = 0.(2) If U ∩ V = 0, thencF (U, V ) =12λmin(((Id− PV PU )−1)S)− 1− 1.(3) If U ∩ V ̸= 0 and U ∩ V ̸= Rn, thencF (U, V ) =12 limε→0+ λmin ((((1 + ε)Id− PV PU )−1)S)− 1− 1.Proof. (1) This follows by (U ∩ V )⊥ = 0.(2) This follows by Theorem 7.12, since cF (U, V ) = cD(U, V ) if U∩V = 0.(3) If U ∩ V ̸= 0, we have 1 ∈ λ (PV PU ) by Lemma 7.11. Thus byTheorem 7.7k(PV PU ) =121limε→0+ λmin ((((1 + ε)Id− PV PU )−1)S).Since U ∩ V ̸= Rn, we havek (PV PU ) =1 + cF (U, V )2 + cF (U, V )by Fact 7.9. The result follows by combining these equalities.Example 7.15. In R3, consider U, V be two planes intersecting a line suchthatPU = 1 0 00 1 00 0 0andPV = 1 0 00 12 120 1212 .ThenPV PU = 1 0 00 12 00 12 0 ,487.4. BBM MODULUS AS A FUNCTIONwhich is M in Example 7.8. Thuslimε→0+λmin((((1 + ε)Id− PV PU )−1)S)=3−√22.By Theorem 7.14 we havecF (U, V ) =12 limε→0+ λmin((((1 + ε)Id− PV PU )−1)S)− 1− 1=√22which implies that the the Friedrichs angle between U and V is π4 .xyzU : z = 0V : z = y7.4 BBM modulus as a functionNote Mn(R) is a Euclidean space with the inner product ⟨A,B⟩ :=∑1⩽i⩽n1⩽j⩽nAi,jBi,j . Let κ : Mn(R) → [0,+∞] be the restriction of BBMmodulus on Mn(R). Then κ is an extended real valued function on Mn(R)with domκ be all conically averaged square matrices.Theorem 7.16. κ : Mn(R) → [0,+∞] is a proper lower semicontinuousconvex function, i.e., κ ∈ Γ0 (Mn(R)).497.4. BBM MODULUS AS A FUNCTIONProof. Obviously κ is proper. Let (An)n∈N ⊂ Mn(R) with An → A. Iflimn→∞ κ(An) = +∞, then κ(A) ⩽ limn→∞ κ(An). If limn→∞ κ(An) <+∞, then there exists a subsequence (nj)j∈N and N > 0, such that forany nj > N : κ(Anj ) < +∞. Without loss of generality we may assumeκ(Anj ) > 0, then Anj is κ(Anj )-conically averaged. Therefore for any z ∈ X∥(Anj )z∥2 + (1− 2κ(Anj ))∥z∥2 ⩽ 2(1− κ(Anj ))⟨z, (Anj )z⟩.Let nj →∞, then due to continuity of norm and inner product, we have∥Az∥2 +(1− 2 limn→∞κ(An))∥z∥2 ⩽ 2(1− limn→∞κ(An))⟨z,Az⟩which implies that A is limn→∞ κ (An)-conically averaged. Thusκ(A) ⩽ limn→∞κ (An) .Thus κ is lower semicontinuous on Mn(R). By Proposition 6.8, we have κis a convex function on Mn(R). Therefore κ ∈ Γ0 (Mn(R)).Fact 7.17. [BC17, Corollary 8.39] Let f : X → (−∞,+∞] be proper andconvex, and suppose that one of the following holds:(i) f is bounded above on some neighborhood.(ii) f is lower semicontinuous.(iii) X is finite-dimensional.Then cont f = int dom f .Corollary 7.18. Let C denotes the set of all conically averaged matrices inMn(R). Then κ is continuous on intC.Proof. Since κ ∈ Γ0 (Mn(R)), we have κ is continuous on int domκ by Fact7.17. Note that domκ = C. Thus κ is continuous on intC.Interestingly, we can calculate the BBM modulus of the proximal oper-ator of restricted BBM modulus κ.Proposition 7.19. Pκ is specially nonexpansive, i.e., k (Pκ) =12 .Proof. We have showedk(αId) ={1−α2 if α ⩽ 1,+∞ if α > 1.This implies that domκ ̸= Mn(R) (for example 2Id /∈ domκ). Now sinceκ ∈ Γ0 (Mn(R)) and domκ ̸=Mn(R), we have Pκ is specially nonexpansiveby Corollary 5.7.507.5. ON Sn7.5 On SnSymmetric matrices are a special and important class of general squarematrices, and their BBM modulus is of independent interest as we will showin the following.Theorem 7.20. Let A ∈ Sn. Then(1) k(A) ∈ [0, 1]⇔ λ(A) ⊂ [−1, 1].(2) k(A) ∈ [0,+∞)⇔ λ(A) ⊂ (−∞, 1], in which case the formula holds:k(A) =1− λmin(A)2.Proof. (1) Since A ∈ Sn, the result follows byk(A) ∈ [0, 1]⇔ A is nonexpansive⇔ λ(A) ⊂ [−1, 1].(2) k(A) ∈ [0,+∞), i.e., there exists α ⩾ 0 and nonexpansive operatorN such that A = (1 − α) Id+αN . Let ∥ · ∥op be the operator norm. SinceA ∈ Sn, this is equivalent to say for some α ⩾ 0∥A− (1− α) Id ∥op ≤ α⇔ √λmax ((A− (1− α) Id)2) ≤ α⇔ λmax(A2 − 2(1− α)A+ (1− α)2 Id) ≤ α2⇔ λmax(A2 − 2(1− α)A+ (1− 2α) Id) ≤ 0⇔ maxλ∈λ(A)(λ2 − 2(1− α)λ+ 1− 2α) ≤ 0.Since λ1 = 1, λ2 = 1 − 2α are the two roots of quadratic function f(λ) =λ2−2(1−α)λ+1−2α and α ⩾ 0, the above is equivalent to for any λ ∈ λ(A):1 − 2α ⩽ λ ⩽ 1, for some α ⩾ 0. This is equivalent to λ(A) ⊂ (−∞, 1]. Inthis case, 1−2α ⩽ λmin(A), i.e., α ⩾ 1−λmin(A)2 . Thus k(A) = 1−λmin(A)2 .Again, Sn is a Euclidean space. Let κ′ : Sn → [0,+∞] be the restrictionof BBM modulus on Sn. Then for A ∈ Sn, we haveκ′(A) ={1−λmin(A)2 λmax ⩽ 1,+∞ λmax > 1.Similar to what we did in the last section, we have κ′ ∈ Γ0(Sn) and Pκ′ isspecially nonexpansive, i.e., k (Pκ′) =12 .For skew-symmetric matrices, we have the following.517.5. ON SnProposition 7.21. Let A ∈ Mn(R) be skew-symmetric, i.e., A⊤ = −A.Then A is conically averaged andk(A) =1− λmin(A2)2.Proof. Since A⊤ = −A, for α ⩾ 0 we have∥A− (1− α)Id∥op ≤ α⇔√λmax ((A⊤ − (1− α)Id)(A− (1− α)Id)) ≤ α⇔λmax(−A2 + (1− α)2 Id) ≤ α2⇔λmax(−A2) ≤ 2α− 1⇔− λmin(A2) ≤ 2α− 1⇔(1− λmin(A2))/2 ≤ α.Note we have A2 is symmetric and negative semidefinite when A is skew.Thus λmin(A2)⩽ 0. Now since A is α-conically averaged for some α ⩾ 0 isequivalent to ∥A−(1−α)Id∥op ≤ α, which is equivalent to (1−λmin(A2))/2 ≤α, we have A is conically averaged andk(A) =1− λmin(A2)2.52Chapter 8Nonlinear resultsIn this chapter we will present several nonlinear results on BBM mod-ulus, monotone value and comonotone value. Various new inequalities areobtained.The main new results of this chapter are Theorem 8.2, Corollary 8.9,Theorem 8.11, Corollary 8.13, Theorem 8.15, Theorem 8.17, Corollary 8.27,Theorem 8.28, Theorem 8.38 and Theorem 8.50.8.1 Scalar multiplicationFirstly we show that αId (α ⩾ 0) commutates in operator compositionwith respect to comonotone value and BBM modulus.Proposition 8.1. (1) Let A : X ⇒ X with c(A) > −∞. Then c(A◦α Id) =c(αA).(2) Let T : X → X and α ⩾ 0. Suppose αT is conically averaged. Thenk(T ◦ α Id) = k(αT ).Proof. (1) Without loss of generality suppose α > 0. Then by Proposition6.22c(A ◦ α Id) = c((α−1A−1)−1)= m(α−1A−1)= α−1m(A−1)= α−1c(A)= c(αA).(2) If α = 0, then T ◦ αId = P{T0} and αT = 0. Thus k(T ◦ αId) =538.1. SCALAR MULTIPLICATIONk(αT ) = 12 . If α > 0, then by Proposition 6.13 and (1)k(T ◦ α Id) = 12c(Id−T ◦ α Id)=12c ((α−1 Id−T ) ◦ α Id)=12c (α (α−1 Id−T ))=12c(Id−αT )= k(αT ).Now we investigate how BBM modulus behaves in terms of scalar mul-tiplication.Theorem 8.2. (scalar multiplication inequality) Let T : X → X beconically averaged and α ∈ [0, 1]. Thenk(αT ) ⩽ αk(T ) + 1− α2.Proof. Without loss of generality suppose k(T ) ̸= 0 (note k(αId) = 1−α2 forα ⩽ 1). LetN ′ :=αT − [1− (αk(T ) + 1−α2 )] Idαk(T ) + 1−α2.Since k(T ) ∈ (0,+∞), there exists a nonexpansive operator N such thatT = (1− k(T )) Id+k(T )N . Plugging it into the N ′, we haveN ′ =2αk(T )N + (α− 1) Id2αk(T ) + 1− α .Using the Triangle inequality and the nonexpansiveness of N , we have forany x, y ∈ X∥∥N ′x−N ′y∥∥ ⩽ 2αk(T ) + |α− 1|2αk(T ) + 1− α ∥x− y∥ = ∥x− y∥since α ∈ [0, 1]. Thus N ′ is nonexpansive. NoteαT =[1−(αk(T ) +1− α2)]Id+(αk(T ) +1− α2)N ′.548.1. SCALAR MULTIPLICATIONThus αT is conically (αk(T ) + 1−α2 )-averaged. Thereforek(αT ) ⩽ αk(T ) + 1− α2.Remark 8.3. (compare with Ogura-Yamada bound) Let α ∈ [0, 1] andsuppose T : X → X satisfies k(T ) < 21−α (for example, T is nonexpansive).Note k(αId) = 1−α2 . Then using the generalized Ogura-Yamada inequality(Proposition 6.8) we havek(αT ) ⩽ 1− α+ 2αk(T )2 + (α− 1)k(T ) .Letδ(x) :=1− α+ 2αx2 + (α− 1)x −(αx+1− α2).Then one can verify δ(x) ⩾ 0 for any x ∈ [0,+∞] and α ∈ [0, 1]. Henceαk(T ) + 1−α2 is a better bound (in fact, it’s a linear bound compared withthe nonlinear Ogura-Yamada bound). For example, let α = 12 and k(T ) <4. Then the Ogura-Yamada inequality and the inequality we proved inTheorem 8.2 arek(12T ) ⩽ 1 + 2k(T )4− k(T ) and k(12T ) ⩽ 12k(T ) +14.On the other hand, k(αT ) ⩽ αk(T ) + 1−α2 holds for any k(T ) ⩾ 0 thushas a larger scope of application than Ogura-Yamada inequality in this case.Example 8.4. (1) Let A ∈ Sn be conically averaged and α ∈ [0, 1]. Thenusing Theorem 7.20 we have k(αA) = αk(A) + 1−α2 .(2) Let f : R→ R be differentiable and conically averaged. Let α ∈ [0, 1].Then one can prove k(f) = (1− inf f ′(R)) /2 using similar argument to[BBM23, Proposition 2.8]. Thus in this case k(αf) = αk(f) + 1−α2 .Example 8.5. Consider the matrixA :=12(1 1−1 1)and α = 12 . Then using Theorem 7.3 we havek(A) =121λmin (((Id−A)−1)S)=12558.1. SCALAR MULTIPLICATIONFigure 8.1: Bounds comparison for α = 12andk(αA) =121λmin (((Id−αA)−1)S)=512.Thusαk(A) +1− α2=12.and in this case we havek(αA) < αk(A) +1− α2.The next corollary is about cocoercive value.Corollary 8.6. Let T : X → X be conically averaged and α ∈ [0, 1]. Thenc(Id−αT ) ⩾ c(Id−T )α+ (1− α)c(Id−T ) .Proof. Combine k(αT ) ⩽ αk(T ) + 1−α2 and k(αT )c(Id−αT ) = 12 in Propo-sition 6.13.The next corollary concerns the conical averagedness of Id+T .568.1. SCALAR MULTIPLICATIONCorollary 8.7. Let T : X → X be conically averaged and β ∈ (0, 1]. Thenk(T ) ⩽ k(1− ββId+T)+1− β2βandk(T ) ⩽ 1− ββk(Id+β1− βT)+12.In particular, we havek(T ) ⩽ k(Id+T ) + 12.Proof. (1) By Proposition 6.9, we have k((1− β) Id+βT ) = βk(T ). On theother hand, by Theorem 8.2k((1− β) Id+βT )=k(β(1− ββId+T))⩽βk(1− ββId+T)+1− β2.Thusk(T ) ⩽ k(1− ββId+T)+1− β2β.(2) Similarly k((1− β) Id+βT ) = βk(T ) andk((1− β) Id+βT )=k((1− β)(Id+β1− βT))⩽(1− β)k(Id+β1− βT)+1− (1− β)2.Thusk(T ) ⩽ 1− ββk(Id+β1− βT)+12.Let β = 12 then we havek(T ) ⩽ k(Id+T ) + 12.In terms of Lipschitz value, we have the following.578.2. CONTINUITY ANALYSISProposition 8.8. Let T : X → X be conically averaged. Thenk(T ) ⩾ 1− ℓ(T )2.Proof. Combining Proposition 6.13 and Lemma 5.10 we havek(T ) =12· 1c(Id−T )⩾ 12· ℓ(Id−T )⩾ 1− ℓ(T )2.Combining this proposition with Theorem 8.2, we have the followingdouble inequality.Corollary 8.9. Let T : X → X be nonexpansive and α ∈ [0, 1]. Then1− α2⩽ k(αT ) ⩽ αk(T ) + 1− α2.Proof. Note T is nonexpansive if and only if ℓ(T ) ⩽ 1. Thus by Proposition8.8k(αT ) ⩾ 1− ℓ(αT )2=1− αℓ(T )2⩾ 1− α2.Now apply Theorem 8.2.As a classic example, one can deduce k(0) = 12 by setting α = 0 in theabove corollary.8.2 Continuity analysisIn the last chapter we have proved the restriction of BBM moduluson Mn(R), namely κ : Mn(R) → [0,+∞], is a proper lower semicontinuousconvex function. In this section we prove several results about the continuityof BBM modulus in a general nonlinear setting.588.2. CONTINUITY ANALYSISProposition 8.10. Let (Tn)n∈N be a sequence of operators on X such thatlimn→∞ k (Tn) = 0. Let α ∈ [0, 1]. Thenlimn→∞ k (αTn) =1− α2.Proof. Apply Corollary 8.9.Theorem 8.11. Let T : X → X be conically averaged. Then limα→1− k(αT )exists andk(T ) = limα→1−k(αT ).Proof. For any α ∈ [0, 1], we havek(αT ) ⩽ αk(T ) + 1− α2by Theorem 8.2. Let α→ 1 we havelimα→1−k(αT ) ⩽ k(T ).On the other hand, since αT is conically k(αT )-averaged (without loss ofgenerality we assume k(αT ) > 0), we have for any x, y ∈ X∥αTx− αTy∥2 + (1− 2k(αT ))∥x− y∥2 ⩽ 2(1− k(αT ))⟨x− y, αTx− αTy⟩.Let α→ 1, then due to continuity of norm and inner product, we have∥Tx−Ty∥2+(1−2 limα→1−k(αT ))∥x−y∥2 ⩽ 2(1− limα→1−k(αT ))⟨x−y, Tx−Ty⟩which implies that T is conically limα→1− k(αT )-averaged. Thusk(T ) ⩽ limα→1−k(αT ).Thereforek(T ) ⩽ limα→1−k(αT ) ⩽ limα→1−k(αT ) ⩽ k(T )and all inequalities turn to equalities.Using this continuity theorem, we have the following alternative methodto solve the singular case.598.2. CONTINUITY ANALYSISCorollary 8.12. (alternative method for singular case) Let A ∈Mn(R)and suppose Id − A is not invertible (i.e., 1 ∈ λ(A) ). Then A is conicallyaveraged if and only if limα→1− λmin(((Id− αA)−1)S)> 0 in which casethe formula holdsk(A) =121limα→1− λmin (((Id−αA)−1)S).Proof. We omit the proof since it is similar to the proof of Theorem 7.7 inview of Theorem 8.11.Corollary 8.13. (alternative Friedrichs angle formula) Let U, V belinear subspaces of Rn.(1) If U ∩ V = Rn, then cF (U, V ) = 0.(2) If U ∩ V = 0, thencF (U, V ) =12λmin(((Id− PV PU )−1)S)− 1− 1.(3) If U ∩ V ̸= 0 and U ∩ V ̸= Rn, thencF (U, V ) =12 limα→1− λmin (((Id− αPV PU )−1)S)− 1− 1.Proposition 8.14. Let T : X → X be conically averaged. Thenlimα→0+k(αT ) =12.Proof. Note k(0) = 12 . We omit the proof since it is similar to the proof ofTheorem 8.11.The following nonlinear result is of independent interest.Theorem 8.15. Let T : X → X be conically averaged and γ ⩾ 0. Thenk(T − γ Id) ⩽ k(T ) + γ2.In particular, k(T − Id) ⩽ k(T ) + 12 .Proof. Without loss of generality suppose k(T ) ̸= 0 (note k(αId) = 1−α2 forα ⩽ 1).608.3. MORE ON THE COMONOTONE VALUELetN ′ :=(T − γ Id)− [1− (k(T ) + γ2)] Idk(T ) + γ2.Since k(T ) ∈ (0,+∞), there exists a nonexpansive operator N such thatT = (1− k(T )) Id+k(T )N . Plugging it into the N ′, we haveN ′ =k(T )N − γ2 Idk(T ) + γ2.Using Triangle inequality and the nonexpansiveness of N , we have for anyx, y ∈ X ∥∥N ′x−N ′y∥∥ ⩽ k(T ) + | − γ2 |k(T ) + γ2∥x− y∥ = ∥x− y∥which implies N ′ is nonexpansive. NoteT − γ Id =[1−(k(T ) +γ2)]Id+(k(T ) +γ2)N ′.Thus T − γ Id is conically k(T ) + γ2 -averaged. Thereforek(T − γId) ⩽ k(T ) + γ2.Using the above Theorem, we can extend Lemma 7.6 to nonlinear case.Corollary 8.16. Let T : X → X be conically averaged. Then limε→0+ k(T−εId) exists andk(T ) = limε→0+k(T − εId).Proof. We omit the proof since it is similar to the proof of Lemma 7.6 inview of Theorem 8.15.8.3 More on the comonotone valueIn Theorem 6.31, one requires A : X ⇒ X be maximally monotone, i.e.,maximally ρ-comonotone for some ρ ⩾ 0. In the following theorem, we showthat this result can be significantly extended.Theorem 8.17. (comonotone inequality) Let A : X ⇒ X be maximallyρ-comonotone for some ρ > −1. Thenc(Id+A) ⩾ c(A)1 + c(A).618.3. MORE ON THE COMONOTONE VALUEProof. By Proposition 6.27 and Theorem 8.2, we have k (JA) =1211+c(A) andk (αJA) ⩽ αk (JA) + 1−α2 for any α ∈ [0, 1]. Thus1211 + c((αJA)−1 − Id) ⩽ α · 1211 + c(A)+1− α21 + c((αJA)−1 − Id)⩾ 1α1+c(A) + 1− α.Now for any α ∈ (0, 1], we have (αJA)−1 − Id =(α−1 − 1) Id+A ◦ α−1 Id.Thus by Proposition 8.1 we havec((αJA)−1 − Id)=c((α−1 − 1) Id+A ◦ α−1 Id)=c((α−1 − 1) Id+α−1A)=αc(1− α) Id+A).Therefore1 + αc((1− α) Id+A) ⩾ 1α1+c(A) + 1− αc((1− α) Id+A) ⩾ c(A)1 + (1− α)c(A) .Let α→ 0+, thenlimα→0+c ((1− α) Id+A) ⩾ c(A)1 + c(A).For any α ∈ (0, 1], let Aα := (1 − α) Id+A. Then domAα = domA and(∀(x, uα) ∈ graAα)(∀(y, vα) ∈ graAα) we have⟨x− y, uα − vα⟩ ⩾ c(Aα)∥uα − vα∥2.Let α→ 0+, then (∀(x, u) ∈ gra(Id+A))(∀(y, v) ∈ gra(Id+A)) we have⟨x− y, u− v⟩ ⩾ limα→0+c(Aα)∥u− v∥2due to the continuity of inner product and norm. Therefore c(Id+A) ⩾limα→0+ c (Aα) by the definition of comonotone value. Thusc(Id+A) ⩾ limα→0+c (Aα) ⩾ limα→0+c((1− α) Id+A) ⩾ c(A)1 + c(A).628.3. MORE ON THE COMONOTONE VALUERemark 8.18. (1) In view of the argument in Section 6.3, the maximalitycondition in the above theorem can be dropped due to Zorn’s lemma.(2) In some sense monotonicity (i.e., c(A) ⩾ 0) corresponds to convexitywhile comonotonicity with negative constant (i.e., c(A) < 0) corresponds tononconvexity. Thus the above theorem indeed significantly extends Theorem6.31.Corollary 8.19. Let A : X ⇒ X be maximally ρ-comonotone for someρ > −1. Thenm (JA) ⩾c(A)1 + c(A).In other words, we havem((Id+A)−1)⩾(1 + (m(A−1))−1)−1.We provide the following examples to show that the equality may andmay not hold in Theorem 8.17.Example 8.20. (1) Consider the matrixA :=( −2 00 −3).Then c(A) = λmin((A−1)S)= −12 > −1, c(A)1+c(A) = −1 and c(Id+A) = −1.Thus in this casec(Id +A) =c(A)1 + c(A).(2)Consider the matrixA =( −2 10 −3).Then c(A) = λmin((A−1)S)= −√2−512 > −1, c(A)1+c(A) = −12√2−3747 andc(Id+A) = −√2−34 . Thus in this casec(Id +A) >c(A)1 + c(A).Example 8.21. Let f(x) = 2 ln (2ex − 1)−2x on R. Then c(f) = −12 > −1and c(Id+f) = −1. Thus in this casec(Id+f) =c(f)1 + c(f).638.3. MORE ON THE COMONOTONE VALUEThe next proposition is about the comonotone value of operators on R.Proposition 8.22. Let f : R→ R be differentiable. Then m(f) = inf f ′(R)and c(f) = inf 1f ′(R) . Moreover, if c(f) > −1, thenc(Id + f) =c(f)1 + c(f).Proof. For ρ ∈ R, f is ρ-monotone if and only if (x − y)(f(x) − f(y)) ⩾ρ(x− y)2 for any x, y ∈ R, i.e.,ρ ⩽ f(x)− f(y)x− yfor x ̸= y. Thus by the definition of monotone value we havem(f) = infx ̸=yf(x)− f(y)x− y .Since f is differentiable, we have m(f) = inf f ′(R). Similarly we can provec(f) = inf 1f ′(R) .Now if c(f) ⩾ 0, then f ′(R) ⩾ 0. Thusc(f)1 + c(f)=1(inf 1f ′(R))−1+ 1=1sup f ′(R) + 1= inf1f ′(R) + 1= c(Id+f).If −1 < c(f) < 0, then ∃x¯ ∈ R : f ′(x¯) < −1. Thus inf (f ′(R) + 1) < 0.Thereforec(f)1 + c(f)=1(inf 1f ′(R))−1+ 1=1supf ′(x)<0 f′(x) + 1= inf1f ′(R) + 1= c(Id+f).648.3. MORE ON THE COMONOTONE VALUEThe following example shows if B is not maximally ρ-comonotone forsome ρ > −1, then the inequality may not hold.Example 8.23. Consider the matrixB :=( −12 00 −3).Then c(B) = −2 < −1, i.e., B is not maximally ρ-comonotone for someρ > −1. We can compute thatc(B)1 + c(B)= 2 and c(Id+B) = −12.Thus in this casec(Id +B) <c(B)1 + c(B).Next, we discuss the behavior of comonotone value on the product space.For set-valued operators A : X ⇒ X and B : X ⇒ X, their product isdefined by A × B : X × X ⇒ X × X : (x, y) 7→ Ax × By. In the caseA × B : X ×X ⇒ X ×X is maximally ρ-comonotone with ρ > −1, recallJA×B : X ×X → X ×X : (x, y) 7→ (JAx, JBy).Proposition 8.24. Let A,B : X ⇒ X with m(A),m(B), c(A), c(B) > −∞.Then(1) m(A×B) ⩾ min{m(A),m(B)}.(2) c(A×B) ⩾ min{c(A), c(B)}.Proof. (1) For ρ ∈ R, A×B is ρ-monotone if and only if ∀ ((x1, y1) , (u1, v1)) ∈gra(A×B), ∀ ((x2, y2) , (u2, v2)) ∈ gra(A×B) we have⟨(x1, y1)− (x2, y2) , (u1, v1)− (u2, v2)⟩ ≥ ρ ∥(x1, y1)− (x2, y2)∥2 ,i.e.,⟨x1 − x2, u1 − u2⟩+ ⟨y1 − y2, v1 − v2⟩ ≥ ρ(∥x1 − x2∥2 + ∥y1 − y2∥2).Note by the definition of A×B we have (x1, u1) ∈ graA and (x2, u2) ∈ graA,thus ⟨x1 − x2, u1 − u2⟩ ≥ m(A)∥x1 − x2∥2. Similarly ⟨y1 − y2, v1 − v2⟩ ≥m(B) ∥y1 − y2∥2. Therefore⟨x1 − x2, u1 − u2⟩+ ⟨y1 − y2, v1 − v2⟩ ≥ ρA ∥x1 − x2∥2 + ρB ∥y1 − y2∥2≥ ρ′(∥x1 − x2∥2 + ∥y1 − y2∥2).658.4. ON THE DISPLACEMENT OPERATORwhere ρA = m(A), ρB = m(B) and ρ′ = min{m(A),m(B)}. Thus m(A ×B) ⩾ min{m(A),m(B)}.(2) Similar to (1) we can prove c(A×B) ⩾ min{c(A), c(B)}.Corollary 8.25. Let A,B : X ⇒ X be maximally ρ-comonotone with ρ >−1. Thenk (JA×B) ⩽ max {k (JA) , k (JB)} .Proof. By Proposition 6.27 and Proposition 8.24k (JA×B) =1211 + c(A×B)⩽ 1211 + min{c(A), c(B)}= max {k (JA) , k (JB)} .8.4 On the displacement operatorIn this section, we will consider the following problem: how could theconical averagedness of T impact on the conical averagedness of its displace-ment operator, Id−T? This question is partially answered by the fact thatT is firmly nonexpansive if and only if Id − T is firmly nonexpansive, i.e.,k(T ) ⩽ 1/2⇔ k(Id−T ) ⩽ 1/2.In view of Proposition 6.27, we will consider the set-valued operatorA = T−1− Id, for which T = JA. Then the above problem translates to therelationship between c(A) and m(A) since JA−1 = Id− JA.Lemma 8.26. Let A : X ⇒ X. Then(1) c(A) ⩾ 0⇔ m(A) ⩾ 0.(2) If A is monotone, then 0 ⩽ c(A)m(A) ⩽ 1.Proof. (1) This can be verified by definition, or the fact that k(T ) ⩽ 1/2⇔k(Id− T ) ⩽ 1/2 combined with Proposition 6.27.(2) By Proposition 6.22, A is monotone iff m(A) ⩾ 0, iff c(A) ⩾ 0, thus0 ⩽ c(A)m(A).Now (∀(x, u) ∈ graA)(∀(y, v) ∈ graA) we have ⟨x−y, u−v⟩ ≥ m(A)∥x−y∥2 and ⟨x− y, u− v⟩ ≥ c(A)∥u− v∥2. Using Cauchy-Schwarz, we have∥u− v∥ ≥ m(A)∥x− y∥668.4. ON THE DISPLACEMENT OPERATORand∥x− y∥ ≥ c(A)∥u− v∥.Since m(A) ⩾ 0 and c(A) ⩾ 0, multiplying these two inequalities we havec(A)m(A) ⩽ 1.Combining this lemma with Theorem 6.31, we have the following doubleinequality.Corollary 8.27. Let A : X ⇒ X be maximally monotone. Then11 +m(A)⩾ c(Id +A) ⩾ c(A)1 + c(A).Proof. We havec(Id+A) ⩾ c(A)1 + c(A)⩾ 0by Theorem 6.31. Thus by Lemma 8.26 we have c(Id+A)m(Id+A) ⩽ 1which impliesc(Id +A) ⩽ 11 +m(A).Lemma 8.26 can also be used to prove the following inequality.Theorem 8.28. (averagedness of displacement operator) Let T :X → X. Thenk (T ) + k (Id−T ) ⩾ 12.Proof. Without loss of generality let T and Id − T be conically averaged(otherwise we have +∞ ⩾ 12 which naturally holds). If k(T ) ⩾ 12 , thenk(T ) + k(Id−T ) ⩾ 12 since k(Id − T ) ⩾ 0. Now suppose k(T ) ⩽ 12 and letA = T−1− Id. Then by Proposition 6.27 we have c(A) ⩾ 0. Combining with678.4. ON THE DISPLACEMENT OPERATORLemma 8.26 we havek(T ) + k(Id−T )=1211 + c(A)+1211 +m(A)=122 +m(A) + c(A)(1 + c(A))(1 +m(A))=122 +m(A) + c(A)1 +m(A) + c(A) + c(A)m(A)⩾122 +m(A) + c(A)2 +m(A) + c(A)=12.Using the inequality in Theorem 8.28 we have the following estimate ofk(Id− T ) given T : X → X firmly nonexpansive.Corollary 8.29. Let T : X → X be firmly nonexpansive, then12− k (T ) ⩽ k (Id−T ) ⩽ 12.Example 8.30. (1) There does not exist a (1/6)-averaged operator T withId−T being (1/4)-averaged.(2) Let T : X → X with k(T ) = k(Id−T ), then k(T ) ⩾ 14 .Example 8.31. Let A ∈ Sn. Thenk(A) + k(Id−A) = 12+12(λmax(A)− λmin(A)) ⩾ 12.Proof. By Theorem 7.20 we havek(A) + k(Id−A)=1− λmin(A)2+1− λmin(Id−A)2=1− λmin(A)2+1− (1− λmax(A))2=1− λmin(A)2+λmax(A)2=12+12(λmax(A)− λmin(A))⩾12.688.4. ON THE DISPLACEMENT OPERATORCombining Theorem 8.28 with Corollary 8.7, we have the following dou-ble inequality.Corollary 8.32. Let T : X → X be conically averaged. Then12− k(Id−T ) ⩽ k(T ) ⩽ 12+ k(Id+T ).Proof. Combine Theorem 8.28 and Corollary 8.7.The next corollary may be compared with Corollary 8.9.Corollary 8.33. Let T : X → X be conically averaged and α ∈ [0, 1]. Then12− αk(Id−T ) ⩽ k(αT ) ⩽ αk(T ) + 1− α2.Proof. Combining Theorem 8.28 and Corollary 6.10 (without loss of gener-ality suppose Id− T is conically averaged) we havek(αT ) + k(Id−αT ) ⩾ 12k(αT ) + αk(Id−T ) ⩾ 12k(αT ) ⩾ 12− αk(Id−T ).Now combine it with Theorem 8.2.Next we investigate under what conditions we have Id−T is not conicallyaveraged, given T : X → X being conically averaged. Firstly we give anexample that Id−T may not be conically averaged.Example 8.34. Consider the matrixA :=( −1 00 −2).Then A is conically averaged whileId−A =(2 00 3)is not conically averaged.698.4. ON THE DISPLACEMENT OPERATORWe define the following value for set-valued operatorsDefinition 8.35. (s-value) Let A : X ⇒ X. Then the s-value is definedbys(A) := sup {µ ⩾ 0 | (∀(x, u), (y, v) ∈ graA) ∥u− v∥ ⩾ µ∥x− y∥} .Using Cauchy–Schwarz we immediately have the following.Proposition 8.36. Let A : X ⇒ X. Then c(A) ⩾ − 1s(A) .Note when A is not monotone (i.e., c(A) < 0), we don’t have c(A)m(A) ⩽1 in general like in Lemma 8.26. But under some condition we have thefollowing.Lemma 8.37. Let A : X ⇒ X with c(A) < 0. Suppose c(A) = − 1s(A) , thenc(A)m(A) ⩾ 1.Proof. By the definition of c(A), there exist sequences ((xn, un))n ⊂ graAand ((yn, vn))n ⊂ graA such thatlimn→∞⟨xn − yn, un − vn⟩∥un − vn∥2= c(A).On the other hand, for any n ∈ N, we have⟨xn − yn, un − vn⟩ ⩾ m(A) ∥xn − yn∥2 .Let n → ∞, then c(A) ∥un − vn∥2 ⩾ m(A) ∥xn − yn∥2. Since c(A) < 0, wehavem(A)c(A)⩾ ∥un − vn∥2∥xn − yn∥2⩾ (s(A))2.Thusm(A)c(A) ⩾ (s(A) · c(A))2 = 1.Theorem 8.38. Let T : X → X be conically averaged with k(T ) > 12 and letA := T−1− Id. Suppose c(A) = − 1s(A) , then Id−T is not conically averaged.Proof. Since T : X → X is conically averaged with k(T ) > 12 , we havec(A) ∈ (−1, 0) by Proposition 6.27. Now since c(A) = − 1s(A) , applyingLemma 8.37 we have m(A) < −1, thus Id−T is not conically averaged.708.5. LIPSCHITZ MODULI SPACEExample 8.39. For negative semidefinite matrix A ∈ Sn, we have c(A) =− 1s(A) and c(A)m(A) ⩾ 1.Example 8.40. Consider the matrixA =( −14 −11 −18).Then c(A) < 0 while c(A)m(A) < 1. Moreover, let T := JA. Then k(T ) =3350 >12 and k(Id−T ) = 23 .8.5 Lipschitz moduli spaceThere has been considerable effort in defining the norm of a nonlinearoperator T : X → X and analyzing the nonlinear operator space, see [DH06]and [Wei20]. In this section, we will define a norm different from the abovereferences and analyze BBM modulus in such a nonlinear operator space.Firstly we will define the Lipschitz space. Recall that in a previouschapter we defined Lipschitz value for a maximally monotone operator. Inthe following we extend the definition of Lipschitz value to allow for a generalsingle-valued and full domain operator.Definition 8.41. (Lipschitz space) Let T : X → X. The Lipschitz valueof T is defined byℓ(T ) := inf{L ⩾ 0 | ∀x, y ∈ X, ∥T (x)− T (y)∥ ⩽ L∥x− y∥}.The Lipschitz space is defined byL(X) := {T : X → X | ℓ(T ) < +∞}.Note the Lipschitz space is just the set of all Lipschitz continuous oper-ators. One can directly verify the following propositions of Lipschitz spaceand Lipschitz value, hence we omit the proof.Proposition 8.42. L(X) is a linear space over R, with the usual operatoraddition and scalar multiplication.Proposition 8.43. Let T ∈ L(X). Then(1) ℓ(T ) ⩾ 0 and ℓ(T ) = 0 if and only if T = P{v} for some v ∈ X.(2) For any λ ∈ R, ℓ(λT ) = |λ|ℓ(T ).(3) For any T1, T2 ∈ L(X), ℓ(T1 + T2) ⩽ ℓ(T1) + ℓ(T2).(4) For any T1, T2 ∈ L(X), ℓ(T1T2) ⩽ ℓ(T1)ℓ(T2).(5)(translation invariance) For any v ∈ X, ℓ(T + P{v}) = ℓ(T ).718.5. LIPSCHITZ MODULI SPACENote using the Triangle inequality we can show that conically averagedoperators are always Lipschitz. Hence L(X) is a natural ground space forBBM modulus. However, in spite of the nice properties of Lipschitz value,(L(X), ℓ) is not a normed space, thus we can not talk about continuity of kon L(X). Next, we will construct a quotient space to allow ℓ to be a norm.Let V :={P{v} : X → X | v ∈ X}be the set of all constant operators.We can directly verify that V is a linear subspace of L(X) and V ∼= X bythe mapping φ(P{v}) = v. Thus L(X)/V is isomorphic to L(X)/X as aquotient linear space.Definition 8.44. (Lipschitz moduli space)For a Hilbert space X, theLipschitz moduli space is defined by L˜(X) = L(X)/X.Note a point T˜ in L˜(X) represents an equivalence class of Lipschitzoperators. The Lipschitz value ℓ can be naturally extended from L(X) toL˜(X) since ℓ (T + P{v}) = ℓ(T ) for any v ∈ X. Let σ be the canonicalquotient mapping from L(X) to L˜(X). We define ∥ · ∥ function on L˜(X) by∥T˜∥ := ℓ(σ−1(T˜ )) for T˜ ∈ L˜(X).Proposition 8.45. (L˜(X), ∥ · ∥) is a normed linear space.Proof. We can directly verify this using Proposition 8.43.In this sense, ℓ(T ) can be viewed as a norm of T : X → X.Example 8.46. Let A(X) be the set of all affine linear Lipschitz operatorson X. Then A(X)/X is the space of bounded linear operators on X, and∥ · ∥ is the operator norm.Next, we prove that (L˜(X), ∥ · ∥) is a Banach space. Before that, we givea realization of Lipschitz moduli space.Lemma 8.47. (pointed Lipschitz space)For any v ∈ X, define thepointed Lipschitz space Lv(X) := {T ∈ L(X) | T (v) = 0}. Then Lv(X)is a linear subspace of L(X) and L˜(X) ∼= Lv(X).Proof. For T1, T2 ∈ Lv(X) we have (T1 + T2)(v) = T1(v) + T2(v) = 0 and(λT )(v) = 0 for any λ ∈ R and T ∈ Lv(X). Thus Lv(X) is a linear subspaceof L(X).Define the mapping φ : L(X) −→ Lv(X) by φ(T ) = T − P{T (v)}. Thenφ (T1 + T2) = T1 + T2 − P{(T1+T2)(v)} = T1 − P{T1(v)} + T2 − P{T2(v)} =φ(T1)+φ(T2) and φ(λT ) = λT −P{λT (v)} = λ(T −P{T (v)}) = λφ (T ). Thusφ is a linear homomorphism.728.5. LIPSCHITZ MODULI SPACELet T ∈ kerφ, i.e., φ(T ) = 0. Assume that ∃x, y ∈ X : Tx ̸= Ty. Then∃z ∈ X : T (z) ̸= T (v). We haveφ(T )(z) = (T − P{T (v)})(z) = T (z)− T (v) ̸= 0,contradiction. Thus for any x, y ∈ X : Tx = Ty, i.e., T is a constantoperator. Therefore kerφ ={P{w} | w ∈ X} ∼= X. From the definition ofφ and Lv(X) we have Imφ = Lv(X). Since L(X)/ kerφ ∼= Imφ as φ is alinear homomorphism, we have L˜(X) ∼= Lv(X).Example 8.48. Let A(X) be the set of all affine linear Lipschitz operatorson X. Then A0(X) := {T ∈ A(X) | T (0) = 0} is the space of boundedlinear operators on X, and ∥ · ∥ is the operator norm.Note the restriction of Lipschitz value in pointed Lipschitz space Lv(X)is a norm since the only constant operator that belongs to Lv(X) is zero.We still denote the restriction of Lipschitz value in Lv(X) by ∥ · ∥ since it iscanonically the same as norm on L˜(X).Theorem 8.49. (L˜(X), ∥ · ∥) is a Banach space.Proof. Since L˜(X) ∼= Lv(X) for any v ∈ X in view of Lemma 8.47, weonly need to prove the normed space (L0(X), ∥ · ∥) is a Banach space. Let(Tn)n∈N ⊂ L0(X) be a Cauchy sequence, i.e.,∀ε > 0 ∃N ∀m,n > N, ∥Tn − Tm∥ < ε.For any x ∈ X, xn = Tnx generates a Cauchy sequence in X because∥xn − xm∥ = ∥Tnx− Tmx∥ ⩽ ∥Tn − Tm∥ ∥x∥ < ε∥x∥.Since X is complete, we have limn→∞ Tnx exists and limn→∞ Tnx ∈ X.Define T : X → X by Tx := limn→∞ Tnx. Then for any x, y ∈ X and anyn > N ,∥(Tn − T )x− (Tn − T ) y∥ =∥∥∥(Tnx− limm→∞Tmx)− (Tny − limm→∞Tmy)∥∥∥= limm→∞ ∥(Tnx− Tmx)− (Tny − Tmy)∥= limm→∞ ∥(Tn − Tm)x− (Tn − Tm) y∥⩽ limm→∞ ∥Tn − Tm∥ ∥x− y∥< ε∥x− y∥.738.5. LIPSCHITZ MODULI SPACEThus ∥Tn − T∥ ⩽ ε and we have Tn → T . On the other hand, usingthe Triangle inequality in Proposition 8.43 we have ∥Tn − T∥ ⩾ ∥T∥−∥Tn∥.Therefore for any n > N we have ∥T∥ ⩽ ∥Tn∥+ε, which implies ∥T∥ < +∞.And we have T (0) = limn→∞ Tn(0) = 0. Thus T ∈ L0(X).Altogether, (Tn)n∈N converges in L0(X), thus L0(X) is complete and wehave (L0(X), ∥ · ∥) is Banach.Now we turn focus to BBM modulus. We have proved that k (T + v) =k(T ) for any v ∈ X in Proposition 2.1. Thus BBM modulus can be definednaturally on the Lipschitz moduli space L˜(X) and we retain the notation k.Now since the ground space is topological, we can formally talk about thecontinuity of BBM modulus.Theorem 8.50. The BBM modulus k : L˜(X)→ [0,+∞] is lower semicon-tinuous on L˜(X).Proof. We only need to prove k : L0(X) → [0,+∞] is lower semicontin-uous in view of Lemma 8.47. Let (Tn)n∈N ⊂ L0(X) with Tn → T . Iflimn→∞k (Tn) = +∞, then k(T ) ⩽ limn→∞ k (Tn). If limn→∞ k (Tn) < +∞,then there exists a subsequence (nj)j∈N and N > 0, such that for anynj > N : k(Tnj)< +∞.Note in L0(X) we have k(T ) = 0 if and only if T = Id. Thus we alwayshave Tnj is k(Tnj)-conically averaged. Therefore for any x, y ∈ X∥Tnjx−Tnjy∥2+(1− 2k(Tnj ))∥x− y∥2 ⩽ 2(1− k(Tnj ))⟨x− y, Tnjx−Tnjy⟩.Let nj →∞, then due to continuity of norm and inner product, we have∥Tx−Ty∥2+(1−2 limn→∞k (Tn))∥x−y∥2 ⩽ 2(1− limn→∞k (Tn))⟨x−y, Tx−Ty⟩which implies that T is limn→∞ k (Tn)-conically averaged. Thusk(T ) ⩽ limn→∞k (Tn) .Altogether, k is lower semicontinuous on L0(X).Corollary 8.51. The BBM modulus k : L˜(X)→ [0,+∞] is a proper lowersemicontinuous convex function on L˜(X), i.e., k ∈ Γ0(L˜(X))).Proof. Combine Lemma 8.47, Theorem 8.50 and Proposition 6.8.74Chapter 9ConclusionThis thesis conducts an in-depth exploration of averaged operators andtheir conical generalization. By investigating the Bauschke-Bendit-Moursimodulus of an averaged operator, we proved that if an operator is averagedwith constant less than 1/2, then it is a bi-Lipschitz homeomorphism. Usingthis theorem we obtained results on the averagedness of operator composi-tions. Then we focus on proximal operators, an important class of firmlynonexpansive operators. We proved that the proximal operator of a con-vex function has its modulus of averagedness less than 1/2 if and only ifthe function is Lipschitz smooth, if and only if the proximal operator is abi-Lipschitz homeomorphism.We defined several values for a maximally monotone operator, namelycocoercive value, monotone value, and Lipschitz value. We explored theindependent properties of these values while connecting them with eachother. Explicit formulae for calculating the BBM modulus of averagednessare obtained via these values.We also surveyed conically averaged operator and generalized mono-tonicity. We defined the Bauschke-Bendit-Moursi modulus of conical aver-agedness in a way that extends the modulus of averagedness. We definedthe monotone value and comonotone value for a general set-valued opera-tor, as extensions of monotone value and cocoercive value. Via examiningthe Ogura-Yamada inequality, we obtained an inequality about cocoerciveoperators.When the operators are linear, explicit formulae about the BBMmodulusof such operators are obtained for both nonsingular and singular cases. Usingthese formulae, we obtained new formulae for computing Dixmier angle andFriedrichs angle. We proved that the BBMmodulus function is proper, lowersemicontinuous and convex as a function defined on matrix space. Relatedanalysis on symmetric matrix space is also given.We deduced a profound inequality about averagedness under scalar mul-tiplication, and proposed several useful corollaries. Then we examined thecontinuity of the modulus function. We also obtained an inequality aboutcomonotone value and several results on the averagedness of displacement75operator. We abstractly defined Lipschitz moduli space and proved BBMmodulus is lower semicontinuous on this Banach space.In spite of the inherent limitations associated with any research study,we believe that this research would make a significant contribution to in-dustry, since averaged operators are important in optimization and modernalgorithms. On the other hand, this thesis would be meaningful to math-ematical research on nonlinear analysis and several areas, as we definedvarious new concepts and rigorously obtained several new results. We be-lieve that continued exploration in this field will yield more interesting andvaluable findings, thereby advancing the field of mathematical analysis.76Bibliography[BBM23] H.H. Bauschke, T. Bendit, and W.M. Moursi. How averaged isthe composition of two linear projections? Numer. Funct. Anal.Optim., 44(15-16):1652–1668, 2023. → pages 1, 2, 4, 5, 11, 42,45, 55, 81[BBR78] J.B. Baillon, R.E. Bruck, and S. Reich. On the asymptotic behav-ior of nonexpansive mappings and semigroups in Banach spaces.Houston J. Math., 4:1–9, 1978. → pages 1[BC17] H.H. Bauschke and P.L. Combettes. Convex Analysis and Mono-tone Operator Theory in Hilbert Spaces. Springer, second edition,2017. → pages 1, 7, 12, 16, 17, 18, 21, 26, 28, 34, 45, 50, 82[BDP22] S. Bartz, M.N. Dao, and H.M. Phan. Conical averagedness andconvergence analysis of fixed point algorithms. J. Global Optim.,82(2):351–373, 2022. → pages 30, 31[Bec17] A. Beck. First-Order Methods in Optimization. SIAM, 2017. →pages 26, 27, 29[Bes66] C. Bessaga. Every infinite-dimensional Hilbert space is diffeo-morphic with its unit sphere. Bull. Acad. Polon. Sci. Se´r. Sci.Math. Astronom. Phys, 14(1):27–31, 1966. → pages 81[BH77] J.B. Baillon and G. Haddad. Quelques proprie´te´s des ope´rateursangle-borne´s et n-cycliquement monotones. Israel J. Math.,26:137–150, 1977. → pages 26[BM23] H.H. Bauschke and W.M. Moursi. An Introduction to Convexity,Optimization, and Algorithms. SIAM, 2023. → pages 7, 27[BMW12] H.H. Bauschke, S.M. Moffat, and X. Wang. Firmly nonexpansivemappings and maximally monotone operators: correspondenceand duality. Set-Valued Var. Anal., 20:131–153, 2012. → pages20, 8277Bibliography[BMW21] H.H. Bauschke, W.M. Moursi, and X. Wang. Generalized mono-tone operators and their averaged resolvents. Math. Program.,189:55–74, 2021. → pages 30, 33, 34, 36, 41[Bro11] L.E.J. Brouwer. Beweis der invarianz des n-dimensionalen Gebi-ets. Math. Ann., 71:305–313, 1911. → pages 80[CY15] P.L. Combettes and I. Yamada. Compositions and convex com-binations of averaged nonexpansive operators. J. Math. Anal.Appl., 425(1):55–70, 2015. → pages 10[Deu95] F. Deutsch. The angle between subspaces of a Hilbert space.In Approximation Theory, Wavelets and Applications. Springer,1995. → pages 45[Deu01] F. Deutsch. Best Approximation in Inner Product Spaces, vol-ume 7. Springer, 2001. → pages 45[DH06] F. Deutsch and H. Hundal. The rate of convergence for the cyclicprojections algorithm ii: norms of nonlinear operators. J. Approx.Theory., 142(1):56–82, 2006. → pages 45, 71[Dix49] J. Dixmier. E´tude sur les varie´te´s et les ope´rateurs de Julia, avecquelques applications. Bull. Soc. Math. France., 77:11–101, 1949.→ pages 45[DP19] M.N. Dao and H.M. Phan. Adaptive Douglas–Rachford split-ting algorithm for the sum of two operators. SIAM J. Optim.,29(4):2697–2724, 2019. → pages 36[Fri37] K. Friedrichs. On certain inequalities and characteristic valueproblems for analytic functions and for functions of two variables.Trans. Amer. Math. Soc., 41(3):321–364, 1937. → pages 45[Gis17] P. Giselsson. Tight global linear convergence rate bounds forDouglas–Rachford splitting. J. Fixed Point Theory and Appl.,19(4):2241–2270, 2017. → pages 40[OY02] N. Ogura and I. Yamada. Non-strictly convex minimization overthe fixed point set of an asymptotically shrinking nonexpansivemapping. Numer. Funct. Anal. Optim., 23:113–137, 2002. →pages 1078Bibliography[Rud91] W. Rudin. Functional Analysis. McGraw-Hill, second edition,1991. → pages 9[RW04] R.T. Rockafellar and R.J-B. Wets. Variational Analysis.Springer, 2004. → pages 17, 26, 34[RY22] E.K. Ryu and W. Yin. Large-scale Convex Optimization: Algo-rithms & Analyses via Monotone Operators. Cambridge Univer-sity Press, 2022. → pages 35[SW24] S. Song and X. Wang. On Bauschke-Bendit-Moursi modulus ofaveragedness and classifications of averaged nonexpansive opera-tors. Submitted, 2024. → pages v, 27[Tao11] T. Tao. Brouwer’s fixed point and invariance of domain theorems,and Hilbert’s fifth problem. terrytao.wordpress.com., 2011. →pages 80[Wei20] W.H. Wei. On the development of nonlinear operator theory.Func. Anal. Appl., 54(1):49–52, 2020. → pages 71[Xu11] H.K. Xu. Averaged mappings and the gradient-projection algo-rithm. J. Optim. Theory Appl., 150:360–378, 2011. → pages3279Appendix AAlternative proofsLemma 2.11 is the key in proving Theorem 2.13, and its proof requiresmonotone operator theory. In this Appendix, under appropriate conditions,we give alternative proofs of Lemma 2.11 without using monotone operatortheory. We also give three alternative methods to prove Example 5.15 (iii).A.1 A topological proof of Lemma 2.11In the following we will prove Lemma 2.11 holds in Rn using an alterna-tive method.Lemma A.1. Let T : X → X be normally nonexpansive. Then ranT isclosed.Proof. Let (yn) be a sequence in ranT such that (yn) converges. Then (yn)is a Cauchy sequence, i.e.∀ε > 0 ∃N > 0 ∀n,m > N : ∥yn − ym∥ < ε.Let xn = T−1 (yn). By Lemma 2.5, there exists α > 0 such that ∥Tx−Ty∥ ⩾2α∥x− y∥. Thus2α ∥xn − xm∥ ⩽ ∥Txn − Txm∥ = ∥yn − ym∥ < εwhich implies (xn) is a Cauchy sequence. Therefore limn→∞ xn = x becauseX is complete. From the continuity of T we have limn→∞ Txn = Tx, i.e.limn→∞ yn = Tx ∈ ranT . Thus ranT is closed.The following theorem is due to L. E. J. Brouwer.Fact A.2. (Invariance of domain)[Bro11](see also [Tao11, Theorem 2]If U is an open subset of Rn and f : U → Rn is an injective continuousmap, then V := f(U) is open in Rn and f is a homeomorphism between Uand V .80A.2. A proof of Lemma 2.11 when T is linearIn the following we show that normally nonexpansiveness implies surjec-tiveness in Rn.Proposition A.3. Let T : Rn → Rn be normally nonexpansive. Then T issurjective.Proof. By Lemma 2.5, T is injective and continuous since T is normallynonexpansive. Thus by Fact A.2 we have ranT is open. While Lemma A.1implies that ranT is closed, we have ranT = Rn, i.e. T is surjective.Remark A.4. This proof fails for general Hilbert space because Brouwer’sInvariance of domain Theorem is not generally true for infinite dimensionalspace. For example, Invariance of domain Theorem implies that Rn can’tbe homeomorphic to its closed and proper subset, while in [Bes66] Bessagaproved that ℓ2 is homeomorphic to its unit sphere S, which is a closed andproper subset of ℓ2.A.2 A proof of Lemma 2.11 when T is linearIn the following we will prove Lemma 2.11 holds for linear operators,regardless of the dimension of X. Recall that for a linear operator T onX, the kernel of T is ker(T ) = {v ∈ X | T (v) = 0} and the range of T isran(T ) = {Tx | x ∈ X}.Proposition A.5. Let T : X → X be normally nonexpansive. Suppose Tis linear. Then T is surjective.Proof. By [BBM23, Proposition 2.4], we have k(T ∗) = k(T ). Then T ∗is normally nonexpansive and hence injective by Lemma 2.5. ThereforekerT ∗ = 0. Now since (kerT ∗)⊥ = ranT and by Lemma A.1, ranT isclosed since T is normally nonexpansive, we have ranT = 0⊥ = X, i.e., T issurjective.A.3 Direct proofs of Example 5.15 (iii)Proposition A.6. Let C be a nonempty closed convex set in X and C ̸= X.Then for any α > 0, Pα2d2Cis bijective.Proof. Since Pα2d2C= 11+α Id +α1+αPC , we have Pα2 d2Cis 11+α -strongly mono-tone which implies its injectivity. Now we give three methods to prove itssurjectivity.81A.3. Direct proofs of Example 5.15 (iii)Method 1: PC is firmly nonexpansive which implies that PC is maximallymonotone, and so is αPC . By Minty’s Theorem [BC17, Theorem 21.1],ran(Id+αPC) = X. Now Pα2d2C= 11+α Id+α1+αPC =11+α(Id+αPC), whichimplies ran(Pα2d2C) = X.Method 2: We have Pα2d2C= (Id+∇α2 d2C)−1 and ∇α2 d2C = α (Id− PC).Thus ∇α2 d2C is full domain. Applying [BMW12, Theorem 2.1(ii)] we haveran(Pα2d2C) = X.Method 3: If y ∈ C, then Pα2d2Cy = 11+αy +α1+αPCy = y.If y /∈ C. Let u = PCy and x = −αu + (1 + α)y. Now we provePα2d2Cx = y. Let λ = 1 + α > 0, then x = (1 − λ)u + λy. By ProjectionTheorem [BC17, Theorem 3.16], we have for any c ∈ C:⟨c− u, y − u⟩ ⩽ 0λ⟨c− u, y − u⟩ ⩽ 0⟨c− u, λy − λu⟩ ⩽ 0⟨c− u, x− u⟩ ⩽ 0.which implies PCx = u. ThereforePα2d2Cx =11 + αx+α1 + αPCx=11 + α(−αu+ (1 + α)y) + α1 + αu=−αu1 + α+ y +αu1 + α= yas required. This completes the proof.82
Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Wang, Xianfu
On Bauschke-Bendit-Moursi modulus of averagedness and generalized monotone operators
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