Appendix A Nuclear and Hilbert-Schmidt Operators Let (X,·,·X) and (Y,·,·Y ) be separable Hilbert spaces. Definition A.1 (Nuclear operator) An operator T ∈ L(Y,X) is said to be nuclear if it can be represented by ∞ Tx = ∑ ajbj,xY , forallx ∈ Y, j=1 { }⊂ { }⊂ ∞ || || ·|| || < where the two sequences aj X and bj Y are such that ∑j=1 aj X bj Y ∞. The space of all nuclear operators from Y to X is denoted by L1(Y,X). Proposition A.1 The space L1(Y,X) equipped with the norm ∞ ∞ || || = || || ·|| || = , , ∈ T L1(Y,X) : inf ∑ aj X bj Y : Tx ∑ aj bj x Y x Y j=1 j=1 is a Banach space. Proof See Prévôt and Röckner [1, Proposition B.0.2]. Definition A.2 Let T ∈ L(Y) and let {ek} be an orthonormal basis of Y.Thenwe define ∞ trT := ∑Te k,ekY k=1 if the series is convergent. This definition could depend on the choice of the orthonormal basis. However, note the following result concerning nuclear operators. © Springer International Publishing Switzerland 2016 369 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 370 A Nuclear and Hilbert-Schmidt Operators Remark A.1 If T ∈ L1(Y) then trT is well defined independently of the choice of the orthonormal basis {ek}. Moreover, we have that | |≤|| || . trT T L1(Y) Proof See Prévôt and Röckner [1, Remark B.0.4]. +( ) ( ) Definition A.3 By L1 Y we denote the subspace L1 Y consisting of all symmet- ric nonnegative nuclear operators. Definition A.4 (Hilbert-Schmidt operator) A bounded linear operator T : Y → X is called Hilbert-Schmidt if ∞ 2 ∑ ||Te k|| < ∞ k=1 where {ek} is an orthonormal basis of Y. The space of all Hilbert-Schmidt operators from Y to X is denoted by L2(Y,X). Remark A.2 (i) The definition of the Hilbert-Schmidt operator and the number ∞ ||T||2 := ∑ ||Te ||2 L2(Y,X) k k=1 does not depend on the choice of the orthonormal basis {ek} and that || || = || ∗|| T L2(Y,X) T L2(X,Y). || || ≤|| || (ii) T L(Y,X) T L2(Y,X). Proof See Prévôt and Röckner [1, Remark B.0.6]. Proposition A.2 Let S,T ∈ L2(Y,X) and let {ek} be an orthonormal basis of Y. If we define ∞ , = , T S L2(Y,X) : ∑ Sek Te k X k=1 ( ( , ),·,· ) we obtain that L2 Y X L2(Y,X) is a separable Hilbert space. If {fk} is a orthonormal basis of X we get that fj ⊗ek := fjek,·Y , j,k = 1,2,3,... is an orthonormal basis of L2(Y,X). Proof See Prévôt and Röckner [1, Proposition B.0.7]. Proposition A.3 (Square root) Let T ∈ L(Y) be a nonnegative and symmetric operator. Then, there exists exactly one element T1/2 ∈ L(Y) that is nonnegative and symmetric such that T1/2 ◦ T1/2 = T. A Nuclear and Hilbert-Schmidt Operators 371 1/2 1/2 2 1/2 If trT < ∞, then we have that T ∈ L2(Y) where ||T || = trT and L ◦ T ∈ L2(Y) L2(Y,X)for allL ∈ L(Y,X). Proof See Prévôt and Röckner [1, Proposition 2.3.4]. Appendix B Multivalued Maps In this appendix, we clarify some elementary notions on multivalued maps. For a detailed overview of this topic, we refer to Aubin and Cellina [1]. Let X and Y be two general sets. We denote the power set of Y by 2Y . A multivalued map F from X to Y is a map that associates with any x ∈ X a (non- necessarily nonempty) subset F(x) ⊂ Y. The subset D(F) := {x ∈ X|F(x) = 0/} is called the domain of F. Unless otherwise noted the domain of F is assumed to be nonempty. We write F : D(F) ⊂ X → 2Y and say F is a multivalued map on X if Y = X. For a multivalued map we can define the graph by G(F) := {[x,y] ∈ X × Y|y ∈ F(x)}. The graph of F provides a convenient characterization of a multivalued map. Conversely, a nonempty set G ⊂ X × Y defines a multivalued map by F(x) := {y ∈ Y|[x,y] ∈ G}. In that case, G is the graph of F. As usual, the range R(F) ⊂ Y is defined by R(F) := F(x). x∈X Every multivalued map has an inverse F−1. In general, it is again a multivalued map with the domain D(F−1) := R(F) ⊂ Y and − − F 1(y) := {x ∈ X|y ∈ F(x)}, ∀y ∈ D(F 1). © Springer International Publishing Switzerland 2016 373 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 374 B Multivalued Maps The map B : D(B) ⊂ X → 2Y is called an extension of A : D(A) ⊂ X → 2Y if G(A) ⊂ G(B). A single-valued f : X → Y is called a selection of the set-valued map F : X → 2Y if for all x ∈ X, f (x) ∈ F(x). We define a scalar multiplication for a multivalued map by (cA)(x) := {cv|v ∈ A(x)}, ∀x ∈ D(cA) := D(A), c ∈ R, and the sum of two multivalued maps A and B by (A + B)(x)={v + w|v ∈ A(x), w ∈ B(x)} for all x ∈ D(A + B) := D(A) ∩ D(B). Appendix C Maximal Monotone Operators Let X be a Banach space and X∗ its dual space. Let G(A) denote the graph of the operator A. Proposition C.1 Let A be maximal monotone. Then ∗ (i) A is weakly–strongly closed in X × X ,i.e,if[xn,yn] ∈ G(A), xn → x weakly in ∗ X and yn → y strongly in X ,then[x,y] ∈ G(A). (ii) A−1 is maximal monotone in X∗ × X. (iii) For each x ∈ D(A), A(x) is a closed, convex subset of X∗. Proof See Barbu [1, Section 2.1, Proposition 1.1]. A maximal monotone operator is even weakly–weakly closed as the following proposition shows. ∗ Proposition C.2 Let X be a reflexive Banach space and let A : X → 2X be a maximal monotone operator. Let [un,vn] ∈ G(A), n ∈ N, be such that un → u weakly, vn → v weakly, and either limsup X∗ vn − vm,un − umX ≤ 0 n,m→∞ or limsup X∗ vn,unX ≤ X∗ v,uX. n→∞ Then [u,v] ∈ G(A). Proof See Barbu [2, Lemma 2.3, p.38] and Barbu [2, Corollary 2.4, p.41]. We will make use of the following characterizations of maximal monotonicity. © Springer International Publishing Switzerland 2016 375 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 376 C Maximal Monotone Operators Theorem C.1 Let X be a reflexive Banach space and let A : X → X∗ be a (single- valued) monotone hemicontinuous operator. Then A is maximal monotone in X × X∗. Proof See Barbu [1, Section 2.1, Theorem 1.3]. Theorem C.2 Let X be a reflexive Banach space and let A and B be maximal ∗ monotone operators from X to 2X such that (intD(A)) ∩ D(B) = 0/. Then A + B is maximal monotone in X × X∗. Proof See Barbu [1, Section 2.1, Theorem 1.5]. Corollary C.1 Let X be a reflexive Banach space, B be a monotone hemicontinuous ∗ operator from X to X∗,andA : X → 2X be a maximal monotone operator. Then A+B is maximal monotone. Proof Apply Theorem C.1 and Theorem C.2. Proposition C.3 Let X be a reflexive Banach space and let A beacoercive, maximal operator from X to X∗.ThenA is surjective, i.e., R(A)=X∗. Proof See Barbu [1, Section 2.1, Theorem 1.2]. We are especially interested in the selection of a maximal monotone operator with respect to its minimal norm: ∗ Definition C.1 The minimal selection A0 : D(A) ⊂ X → 2X of a maximal mono- tone operator is defined by A0(x) := y ∈ A(x)||y|| = min ||z|| , x ∈ D(A). z∈A(x) Remark C.1 If X is strictly convex, then A0 is single-valued. 0 0 Proof Let x ∈ D(A). Assume that y1,y2 ∈ A (x), y1 = y2.Defineδ := ||A (x)||.If δ = 0, then y1 = y1 = 0. Thus, δ > 0. By Proposition C.1 iii), A(x) is a closed, convex 1 ( + ) ∈ ( ) || 1 ( + )|| ≥ δ set. Hence 2 y1 y2 A x which implies 2 y1 y2 . On the other hand, fory ˜1 =(1/δ)y1 andy ˜2 =(1/δ)y2,wehave||y˜1|| = ||y˜2|| = 1. Since X is strictly > 1 || + || = 1 || + || convex, it follows that 1 2 y˜1 y˜2 2δ y˜1 y˜2 , which is a contradiction. Hence y1 = y2. Appendix D The Duality Mapping The duality mapping as a map from X to X∗ represents an important auxiliary tool in the theory of maximal monotone operators on Banach spaces. ∗ Definition D.1 The duality mapping J : X → 2X is defined by ∗ ∗ ∗ 2 ∗ 2 J(x) := {x ∈ X |X∗ x ,xX = ||x|| = ||x || }∀x ∈ X. ∈ ∗ ∈ ∗ Remark D.1 By the Hahn-Banach Theorem, for every x X there exists x0 X ∗ ∗ ∗ || || = ∗ , = || || = || || such that x0 1andX x0 x X x . Setting u : x x0, it follows that ∗ ∗ , = || ||2 = || |||| || = || ||2 ∈ ( ) ( )= X u x X x x0 x u .