- Email: [email protected]
Smooth points in spaces of operators
Linear Algebra and its Applications 517 (2017) 129–133
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Smooth points in spaces of operators T.S.S.R.K. Rao Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India
a r t i c l e
i n f o
Article history: Received 4 November 2016 Accepted 9 December 2016 Available online 14 December 2016 Submitted by P. Semrl MSC: primary 47L05, 46B20
a b s t r a c t For Banach spaces X, Y , in the space of bounded linear operators L(X, Y ), we examine the relation between T ∈ L(X, Y ) being a smooth point versus T ∗ ∈ L(Y ∗ , X ∗ ) being a smooth point. Motivated by some results in the recent paper of Paul et al., we give some sufficient conditions for the validity of such a statement for spaces of operators. © 2016 Elsevier Inc. All rights reserved.
Keywords: Smooth points Spaces of operators Hahn–Banach smooth spaces
1. Introduction Let X be a real Banach space. A non-zero vector x0 ∈ X is said to be a smooth point, if there is a unique unit vector x∗ ∈ X ∗ such that x∗ (x0 ) = x0 . A well-known theorem of Mazur asserts that in a separable Banach space smooth points are dense (see [2] page 171). An interesting question is to study these points for L(X, Y ) and its subspaces. We use standard linear algebra of tensors and basic functional analysis in our analysis. By an operator we mean a bounded linear map. We recall that x ∈ X is said to be Birkhoff–James orthogonal to y ∈ X if x ≤ x + λy for all real numbers λ. It is known that x0 is a smooth point if and only if E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.laa.2016.12.011 0024-3795/© 2016 Elsevier Inc. All rights reserved.
130
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
whenever x0 is orthogonal to y and x0 is orthogonal to z implies x0 is orthogonal to y + z. In [3], the authors relate, in some special cases, the orthogonality of operators in K(X, Y ) to the orthogonality of vectors in Y at the points where the operators attain the norm. Since T → T ∗ is an isometry (surjective only when Y is reflexive, see [2] Exercise 3.9 on page 192), it is an interesting question to consider when smoothness gets preserved in K(Y ∗ , X ∗ ) under this into isometry. Throughout the paper we consider a Banach space X as canonically embedded in its bidual and recall that for a continuous linear map T on X, T ∗∗ = T on X. We first note the easy implication that if T ∗ is a smooth point of K(Y ∗ , X ∗ ), then since it is also a smooth point of {S ∗ : S ∈ K(X, Y )}, T is a smooth point of K(X, Y ). For x∗∗ ∈ X ∗∗ and y ∗ ∈ Y ∗ , we denote by x∗∗ ⊗ y ∗ the linear functional defined on the space of operators by (x∗∗ ⊗ y ∗ )(T ) = x∗∗ (T ∗ (y ∗ )), for any operator T and note that x∗∗ ⊗ y ∗ = x∗∗ y ∗ . We denote a rank one operator by x∗ ⊗ y, so that (x∗∗ ⊗ y ∗ )(x∗ ⊗ y) = x∗∗ (x∗ )y ∗ (y). Let y0 ∈ Y be a unit vector and a smooth point. Suppose T = x∗ ⊗ y0 is a smooth point of K(X, Y ), then if T ∗ = y0 ⊗ x∗ is a smooth point of K(Y ∗ , X ∗ ), it is easy to see that y0 is a smooth point of Y ∗∗ . Such points are called, very smooth points in the literature, see [6]. Thus one needs additional conditions to deduce the smoothness of T ∗ from that of T , even for rank one operators. This affects Corollary 4.2.1 in [3]. See also [4]. An interesting question that arises in this context is to give sufficient conditions on X or Y to ensure that T ∈ K(X, Y ) (T ∈ L(X, Y )) is a smooth point implies T ∗ ∈ K(Y ∗ , X ∗ ) (T ∗ ∈ L(Y ∗ , X ∗ )) is a smooth point. We recall that X is said to be a Hahn–Banach smooth space, if under the canonical embedding, functionals in X ∗ have unique norm-preserving extension in X ∗∗∗ . See [1] for several examples of such spaces. We show that if Y is a Hahn–Banach smooth space and T ∈ K(X, Y ) is a smooth point, then T ∗ is a smooth point of K(Y ∗ , X ∗ ). We prove a local version of the above result by showing that if T is a very smooth point of K(X, Y ), then T ∗ is a smooth point of K(Y ∗ , X ∗ ). To answer this question for L(X, Y ), we need stronger geometric assumptions. For a Banach space X, let X1 denote the closed unit ball and let ∂e X1 denote the set of extreme points of X1 . We recall from Chapter I of [1] that a closed subspace M ⊂ X is said to be an M -ideal, if there is a linear projection P : X ∗ → X ∗ such that ker(P ) = M ⊥ and x∗ = P (x∗ ) + x∗ − P (x∗ ) for all x∗ ∈ X ∗ . This geometric condition implies that ∂e X1∗ = ∂e M1∗ ∪ ∂e M1⊥ . Also functionals in M ∗ have unique norm preserving extension in X ∗ . The monograph [1] also studies spaces X, Y for which K(X, Y ) is an M -ideal in L(X, Y ). Let Y be a Hahn–Banach smooth space. Suppose K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ). Let T ∈ L(X, Y ) be a smooth point that attains its norm. We show that if d(T, K(X, Y )) < T , then T ∗ is a smooth point of L(Y ∗ , X ∗ ). Thanks are due to the referee for several suggestions that led to this improved version.
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
131
2. Main results We recall from [5] that ∂e K(X, Y )∗1 = {x∗∗ ⊗ y ∗ : x∗∗ ∈ ∂e X1∗∗ , y ∗ ∈ ∂e Y1∗ }. Theorem 1. Let Y be a Hahn–Banach smooth space. For any smooth point T ∈ K(X, Y ), T ∗ is a smooth point of K(Y ∗ , X ∗ ). Proof. Let T ∈ K(X, Y ) be a smooth point and suppose T = 1. Since T ∗ is compact, ∗∗ ∗∗ ∗ ∗ ∗∗ let y0∗ = 1 = T ∗ (y0∗ ). Let x∗∗ 0 ∈ ∂e X1 be such that x0 (T (y0 )) = 1. Thus (x0 ⊗ y0∗ )(T ) = 1. Since T is smooth, it is easy to see that y0∗ ∈ ∂e Y1∗ . As Y is Hahn–Banach smooth, y0∗ ∈ ∂e Y1∗∗∗ . Let K = {Λ ∈ K(Y ∗ , X ∗ )∗1 : Λ(T ∗ ) = 1}. K equipped with the weak∗ -topology is a compact, convex, extreme subset of K(Y ∗ , X ∗ )1 . We have from the result of Ruess ∗ and Stegall quoted above that y0∗ ⊗ x∗∗ 0 ∈ ∂e K. Clearly T is smooth if and only if K ∗ ∗∗ is the singleton {y0 ⊗ x0 }. Suppose K is not a singleton. Then by the Krein–Milman theorem, as K is an extreme subset of the dual unit ball, there is another extreme functional, which by the result of Ruess and Stegall, again will be of the form, τ ⊗ x∗∗ , with τ (T ∗∗ (x∗∗ )) = 1, for some τ ∈ ∂e Y1∗∗∗ and x∗∗ ∈ ∂e X1∗∗ . Let y ∗ = τ |Y . We now consider the functional x∗∗ ⊗ y ∗ ∈ K(X, Y )∗1 . As T is a compact operator we note that T ∗∗ maps X ∗∗ into Y . Thus (x∗∗ ⊗ y ∗ )(T ) = 1 and hence x∗∗ ⊗ y ∗ = 1 = y ∗ . Since ∗ ∗ ∗ T is smooth, x∗∗ ⊗ y ∗ = x∗∗ 0 ⊗ y0 on K(X, Y ). If y and y0 are linearly independent ∗ ∗ ∗ on Y , let y0 be such that y (y0 ) = 1 and y0 (y0 ) = 0. Fix x ∈ X ∗ such that x∗∗ (x∗ ) = 1. Now consider the rank one operator S = x∗ ⊗ y0 ∈ K(X, Y ). We have (x∗∗ ⊗ y ∗ )(S) = 1, ∗ ∗ ∗ whereas (x∗∗ 0 ⊗ y0 )(S) = 0. This contradiction shows that y = αy0 and |α| = 1. By ∗∗ ∗∗ replacing x0 by −x0 , if necessary, we assume without loss of generality that α = 1. Again as Y is Hahn–Banach smooth, we have, τ = y0∗ . Similarly using linear dependence, ∗ ∗ ∗ one can also show that x∗∗ = x∗∗ 0 . Hence T is a smooth point of K(Y , X ). 2 In the next proposition, we give a local condition, to ensure that smoothness of T implies that of T ∗ . We recall from [6] that a smooth point x ∈ X is said to be very smooth, if x continues to be a smooth point of X ∗∗ . In this case the x∗ ∈ ∂e X1∗ with x∗ (x) = x, is a point of weak∗ –weak continuity for the identity map on X1∗ , see Lemma III.2.14 in [1]. Proposition 2. Let T ∈ K(X, Y ) be a very smooth point. Then T attains its norm and T ∗ is a smooth point of K(Y ∗ , X ∗ ). Proof. Let T = 1 and as in the proof of Theorem 1, let x∗∗ ∈ ∂e X1∗∗ , y ∗ ∈ ∂e Y1∗ be such that (x∗∗ ⊗y ∗ )(T ) = 1. Since T is a very smooth point, by what we have remarked above, x∗∗ ⊗ y ∗ is a point of weak∗ –weak continuity for the identity map on K(X, Y )∗1 . We next claim that x∗∗ is a point of weak∗ –weak continuity for the identity map on X1∗∗ . Since X1 is weak∗ -dense in X1∗∗ , this in particular shows that x∗∗ = x ∈ X1 and thus T attains
132
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
∗∗ ∗∗ ∗∗ its norm. Let {x∗∗ in the weak∗ -topology. For α }α∈Δ ⊂ X1 be a net such that xα → x ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ any S ∈ K(X, Y ), we have (x∗∗ ⊗ y ∗ )(S). α ⊗ y )(S) = xα (S (y )) → x (S (y )) = (x ∗∗ ∗ ∗ ∗ Thus the net {xα ⊗ y }α∈Δ ⊂ K(X, Y )1 , converges in the weak -topology to x∗∗ ⊗ y ∗ . Therefore by our hypothesis, we get that this convergence is also in the weak topology. We next note that the injective tensor product, X ∗ ⊗ˆ Y ⊂ K(X, Y ) and thus by duality, (X ∗ ⊗ˆ Y )∗∗ ⊂ K(X, Y )∗∗ . It is easy to see that X ∗∗∗ ⊗ˆ Y ⊂ (X ∗ ⊗ˆ Y )∗∗ such that under the canonical embedding, (τ ⊗ y)(x∗∗ ⊗ y ∗ ) = τ (x∗∗ )y ∗ (y), for τ ∈ X ∗∗∗ , y ∈ Y . See ∗∗ page 265 of [1]. This implies that x∗∗ in the weak topology of X ∗∗ . Thus x∗∗ is a α →x ∗ point of weak –weak continuity. Similarly one can show that y ∗ is a point of weak∗ –weak continuity for the identity map on Y1∗ . By using Lemma III.2.14 of [1] again, we get that y ∗ has unique norm preserving extension in Y ∗∗∗ . Proceeding as in the proof of Theorem 1, we see that T ∗ is a smooth point of K(Y ∗ , X ∗ ). 2
We next study smooth points of L(Y ∗ , X ∗ ). Theorem 3. Let Y be a Hahn–Banach smooth space. Suppose K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ). For any smooth point T ∈ L(X, Y ) that attains its norm and such that d(T, K(X, Y )) < T , T ∗ is a smooth point of L(Y ∗ , X ∗ ). Proof. As in the proof of Theorem 1, let T = 1 and let K = {Λ ∈ L(Y ∗ , X ∗ )∗1 : Λ(T ∗ ) = 1}. Let K be equipped with the weak∗ -topology. Let x0 = 1 = T (x0 ) and let y0∗ ∈ ∂e Y1∗ be such that y0∗ (T (x0 )) = 1. Since T is a smooth point of L(X, Y ), it is easy to see that T (x0 ) is a smooth point of Y . As K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ), by the properties of M -ideals recalled in the introduction, x0 ⊗ y0∗ ∈ ∂e K. Proceeding as in the proof of Theorem 1, suppose there is a Λ0 ∈ ∂e K such that Λ0 (T ∗ ) = 1. Again by the M -ideal assumption, we have two possibilities. Suppose Λ0 ∈ K(Y ∗ , X ∗ )⊥ . Then Λ0 (T ∗ ) = 1 = d(T ∗ , K(Y ∗ , X ∗ )) ≤ d(T, K(X, Y )) < 1. Thus we only have the possibility, Λ0 ∈ ∂e K(Y ∗ , X ∗ )∗1 . Hence Ruess and Stegall’s theorem ensures that Λ0 = τ ⊗ x∗∗ , as in the proof of Theorem 1. Since in the proof of Theorem 1, we are only manipulating with rank one operators, the conclusion follows. 2 Remark 4. It is clear from the above proof that we only need to assume, extreme points of Y1∗ which attain their norm at a smooth point of Y , have unique norm-preserving extension in Y ∗∗∗ . This assumption is equivalent to the assumption, every smooth point in Y is a very smooth point. Remark 5. We note that if the T in the above theorem is a compact operator and smooth point only in K(X, Y ), then by Theorem 1, we have that T ∗ ∈ K(Y ∗ , X ∗ ) is a smooth point. Since K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ), it is easy to see that T ∗ is a smooth point of L(Y ∗ , X ∗ ). In particular T is also a smooth point of L(X, Y ). It is also easy to see that the same proof goes through, if we assume T ∗ attains its norm and d(T ∗ , K(Y ∗ , X ∗ )) < T . We recall that by Proposition 4 in Section 4 of [7], one has that {T ∈ L(X, Y ) : T ∗ attains its norm} is a dense subset of L(X, Y ).
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
133
References [1] P. Harmand, D. Werner, W. Werner, M -Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993, viii+387 pp. [2] R.B. Holmes, Geometric Functional Analysis and Its Applications, Graduate Texts in Mathematics, vol. 24, Springer-Verlag, New York–Heidelberg, 1975, x+246 pp. [3] K. Paul, D. Sain, P. Ghosh, Birkhoff–James orthogonality and smoothness of bounded linear operators, Linear Algebra Appl. 506 (2016) 551–563. [4] T.S.S.R.K. Rao, Review of ‘Birkhoff–James orthogonality and smoothness of bounded linear operators’, Zbl 06617602, 2016. [5] W. Ruess, C. Stegall, Extreme points in duals of operator spaces, Math. Ann. 261 (1982) 535–546. [6] F. Sullivan, Geometric properties determined by the higher duals of a Banach space, Illinois J. Math. 21 (1977) 315–331. [7] V. Zizler, On some extremal problems in Banach spaces, Math. Scand. 32 (1973) 214–224 (1974).
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Smooth points in spaces of operators T.S.S.R.K. Rao Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India
a r t i c l e
i n f o
Article history: Received 4 November 2016 Accepted 9 December 2016 Available online 14 December 2016 Submitted by P. Semrl MSC: primary 47L05, 46B20
a b s t r a c t For Banach spaces X, Y , in the space of bounded linear operators L(X, Y ), we examine the relation between T ∈ L(X, Y ) being a smooth point versus T ∗ ∈ L(Y ∗ , X ∗ ) being a smooth point. Motivated by some results in the recent paper of Paul et al., we give some sufficient conditions for the validity of such a statement for spaces of operators. © 2016 Elsevier Inc. All rights reserved.
Keywords: Smooth points Spaces of operators Hahn–Banach smooth spaces
1. Introduction Let X be a real Banach space. A non-zero vector x0 ∈ X is said to be a smooth point, if there is a unique unit vector x∗ ∈ X ∗ such that x∗ (x0 ) = x0 . A well-known theorem of Mazur asserts that in a separable Banach space smooth points are dense (see [2] page 171). An interesting question is to study these points for L(X, Y ) and its subspaces. We use standard linear algebra of tensors and basic functional analysis in our analysis. By an operator we mean a bounded linear map. We recall that x ∈ X is said to be Birkhoff–James orthogonal to y ∈ X if x ≤ x + λy for all real numbers λ. It is known that x0 is a smooth point if and only if E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.laa.2016.12.011 0024-3795/© 2016 Elsevier Inc. All rights reserved.
130
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
whenever x0 is orthogonal to y and x0 is orthogonal to z implies x0 is orthogonal to y + z. In [3], the authors relate, in some special cases, the orthogonality of operators in K(X, Y ) to the orthogonality of vectors in Y at the points where the operators attain the norm. Since T → T ∗ is an isometry (surjective only when Y is reflexive, see [2] Exercise 3.9 on page 192), it is an interesting question to consider when smoothness gets preserved in K(Y ∗ , X ∗ ) under this into isometry. Throughout the paper we consider a Banach space X as canonically embedded in its bidual and recall that for a continuous linear map T on X, T ∗∗ = T on X. We first note the easy implication that if T ∗ is a smooth point of K(Y ∗ , X ∗ ), then since it is also a smooth point of {S ∗ : S ∈ K(X, Y )}, T is a smooth point of K(X, Y ). For x∗∗ ∈ X ∗∗ and y ∗ ∈ Y ∗ , we denote by x∗∗ ⊗ y ∗ the linear functional defined on the space of operators by (x∗∗ ⊗ y ∗ )(T ) = x∗∗ (T ∗ (y ∗ )), for any operator T and note that x∗∗ ⊗ y ∗ = x∗∗ y ∗ . We denote a rank one operator by x∗ ⊗ y, so that (x∗∗ ⊗ y ∗ )(x∗ ⊗ y) = x∗∗ (x∗ )y ∗ (y). Let y0 ∈ Y be a unit vector and a smooth point. Suppose T = x∗ ⊗ y0 is a smooth point of K(X, Y ), then if T ∗ = y0 ⊗ x∗ is a smooth point of K(Y ∗ , X ∗ ), it is easy to see that y0 is a smooth point of Y ∗∗ . Such points are called, very smooth points in the literature, see [6]. Thus one needs additional conditions to deduce the smoothness of T ∗ from that of T , even for rank one operators. This affects Corollary 4.2.1 in [3]. See also [4]. An interesting question that arises in this context is to give sufficient conditions on X or Y to ensure that T ∈ K(X, Y ) (T ∈ L(X, Y )) is a smooth point implies T ∗ ∈ K(Y ∗ , X ∗ ) (T ∗ ∈ L(Y ∗ , X ∗ )) is a smooth point. We recall that X is said to be a Hahn–Banach smooth space, if under the canonical embedding, functionals in X ∗ have unique norm-preserving extension in X ∗∗∗ . See [1] for several examples of such spaces. We show that if Y is a Hahn–Banach smooth space and T ∈ K(X, Y ) is a smooth point, then T ∗ is a smooth point of K(Y ∗ , X ∗ ). We prove a local version of the above result by showing that if T is a very smooth point of K(X, Y ), then T ∗ is a smooth point of K(Y ∗ , X ∗ ). To answer this question for L(X, Y ), we need stronger geometric assumptions. For a Banach space X, let X1 denote the closed unit ball and let ∂e X1 denote the set of extreme points of X1 . We recall from Chapter I of [1] that a closed subspace M ⊂ X is said to be an M -ideal, if there is a linear projection P : X ∗ → X ∗ such that ker(P ) = M ⊥ and x∗ = P (x∗ ) + x∗ − P (x∗ ) for all x∗ ∈ X ∗ . This geometric condition implies that ∂e X1∗ = ∂e M1∗ ∪ ∂e M1⊥ . Also functionals in M ∗ have unique norm preserving extension in X ∗ . The monograph [1] also studies spaces X, Y for which K(X, Y ) is an M -ideal in L(X, Y ). Let Y be a Hahn–Banach smooth space. Suppose K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ). Let T ∈ L(X, Y ) be a smooth point that attains its norm. We show that if d(T, K(X, Y )) < T , then T ∗ is a smooth point of L(Y ∗ , X ∗ ). Thanks are due to the referee for several suggestions that led to this improved version.
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
131
2. Main results We recall from [5] that ∂e K(X, Y )∗1 = {x∗∗ ⊗ y ∗ : x∗∗ ∈ ∂e X1∗∗ , y ∗ ∈ ∂e Y1∗ }. Theorem 1. Let Y be a Hahn–Banach smooth space. For any smooth point T ∈ K(X, Y ), T ∗ is a smooth point of K(Y ∗ , X ∗ ). Proof. Let T ∈ K(X, Y ) be a smooth point and suppose T = 1. Since T ∗ is compact, ∗∗ ∗∗ ∗ ∗ ∗∗ let y0∗ = 1 = T ∗ (y0∗ ). Let x∗∗ 0 ∈ ∂e X1 be such that x0 (T (y0 )) = 1. Thus (x0 ⊗ y0∗ )(T ) = 1. Since T is smooth, it is easy to see that y0∗ ∈ ∂e Y1∗ . As Y is Hahn–Banach smooth, y0∗ ∈ ∂e Y1∗∗∗ . Let K = {Λ ∈ K(Y ∗ , X ∗ )∗1 : Λ(T ∗ ) = 1}. K equipped with the weak∗ -topology is a compact, convex, extreme subset of K(Y ∗ , X ∗ )1 . We have from the result of Ruess ∗ and Stegall quoted above that y0∗ ⊗ x∗∗ 0 ∈ ∂e K. Clearly T is smooth if and only if K ∗ ∗∗ is the singleton {y0 ⊗ x0 }. Suppose K is not a singleton. Then by the Krein–Milman theorem, as K is an extreme subset of the dual unit ball, there is another extreme functional, which by the result of Ruess and Stegall, again will be of the form, τ ⊗ x∗∗ , with τ (T ∗∗ (x∗∗ )) = 1, for some τ ∈ ∂e Y1∗∗∗ and x∗∗ ∈ ∂e X1∗∗ . Let y ∗ = τ |Y . We now consider the functional x∗∗ ⊗ y ∗ ∈ K(X, Y )∗1 . As T is a compact operator we note that T ∗∗ maps X ∗∗ into Y . Thus (x∗∗ ⊗ y ∗ )(T ) = 1 and hence x∗∗ ⊗ y ∗ = 1 = y ∗ . Since ∗ ∗ ∗ T is smooth, x∗∗ ⊗ y ∗ = x∗∗ 0 ⊗ y0 on K(X, Y ). If y and y0 are linearly independent ∗ ∗ ∗ on Y , let y0 be such that y (y0 ) = 1 and y0 (y0 ) = 0. Fix x ∈ X ∗ such that x∗∗ (x∗ ) = 1. Now consider the rank one operator S = x∗ ⊗ y0 ∈ K(X, Y ). We have (x∗∗ ⊗ y ∗ )(S) = 1, ∗ ∗ ∗ whereas (x∗∗ 0 ⊗ y0 )(S) = 0. This contradiction shows that y = αy0 and |α| = 1. By ∗∗ ∗∗ replacing x0 by −x0 , if necessary, we assume without loss of generality that α = 1. Again as Y is Hahn–Banach smooth, we have, τ = y0∗ . Similarly using linear dependence, ∗ ∗ ∗ one can also show that x∗∗ = x∗∗ 0 . Hence T is a smooth point of K(Y , X ). 2 In the next proposition, we give a local condition, to ensure that smoothness of T implies that of T ∗ . We recall from [6] that a smooth point x ∈ X is said to be very smooth, if x continues to be a smooth point of X ∗∗ . In this case the x∗ ∈ ∂e X1∗ with x∗ (x) = x, is a point of weak∗ –weak continuity for the identity map on X1∗ , see Lemma III.2.14 in [1]. Proposition 2. Let T ∈ K(X, Y ) be a very smooth point. Then T attains its norm and T ∗ is a smooth point of K(Y ∗ , X ∗ ). Proof. Let T = 1 and as in the proof of Theorem 1, let x∗∗ ∈ ∂e X1∗∗ , y ∗ ∈ ∂e Y1∗ be such that (x∗∗ ⊗y ∗ )(T ) = 1. Since T is a very smooth point, by what we have remarked above, x∗∗ ⊗ y ∗ is a point of weak∗ –weak continuity for the identity map on K(X, Y )∗1 . We next claim that x∗∗ is a point of weak∗ –weak continuity for the identity map on X1∗∗ . Since X1 is weak∗ -dense in X1∗∗ , this in particular shows that x∗∗ = x ∈ X1 and thus T attains
132
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
∗∗ ∗∗ ∗∗ its norm. Let {x∗∗ in the weak∗ -topology. For α }α∈Δ ⊂ X1 be a net such that xα → x ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ any S ∈ K(X, Y ), we have (x∗∗ ⊗ y ∗ )(S). α ⊗ y )(S) = xα (S (y )) → x (S (y )) = (x ∗∗ ∗ ∗ ∗ Thus the net {xα ⊗ y }α∈Δ ⊂ K(X, Y )1 , converges in the weak -topology to x∗∗ ⊗ y ∗ . Therefore by our hypothesis, we get that this convergence is also in the weak topology. We next note that the injective tensor product, X ∗ ⊗ˆ Y ⊂ K(X, Y ) and thus by duality, (X ∗ ⊗ˆ Y )∗∗ ⊂ K(X, Y )∗∗ . It is easy to see that X ∗∗∗ ⊗ˆ Y ⊂ (X ∗ ⊗ˆ Y )∗∗ such that under the canonical embedding, (τ ⊗ y)(x∗∗ ⊗ y ∗ ) = τ (x∗∗ )y ∗ (y), for τ ∈ X ∗∗∗ , y ∈ Y . See ∗∗ page 265 of [1]. This implies that x∗∗ in the weak topology of X ∗∗ . Thus x∗∗ is a α →x ∗ point of weak –weak continuity. Similarly one can show that y ∗ is a point of weak∗ –weak continuity for the identity map on Y1∗ . By using Lemma III.2.14 of [1] again, we get that y ∗ has unique norm preserving extension in Y ∗∗∗ . Proceeding as in the proof of Theorem 1, we see that T ∗ is a smooth point of K(Y ∗ , X ∗ ). 2
We next study smooth points of L(Y ∗ , X ∗ ). Theorem 3. Let Y be a Hahn–Banach smooth space. Suppose K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ). For any smooth point T ∈ L(X, Y ) that attains its norm and such that d(T, K(X, Y )) < T , T ∗ is a smooth point of L(Y ∗ , X ∗ ). Proof. As in the proof of Theorem 1, let T = 1 and let K = {Λ ∈ L(Y ∗ , X ∗ )∗1 : Λ(T ∗ ) = 1}. Let K be equipped with the weak∗ -topology. Let x0 = 1 = T (x0 ) and let y0∗ ∈ ∂e Y1∗ be such that y0∗ (T (x0 )) = 1. Since T is a smooth point of L(X, Y ), it is easy to see that T (x0 ) is a smooth point of Y . As K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ), by the properties of M -ideals recalled in the introduction, x0 ⊗ y0∗ ∈ ∂e K. Proceeding as in the proof of Theorem 1, suppose there is a Λ0 ∈ ∂e K such that Λ0 (T ∗ ) = 1. Again by the M -ideal assumption, we have two possibilities. Suppose Λ0 ∈ K(Y ∗ , X ∗ )⊥ . Then Λ0 (T ∗ ) = 1 = d(T ∗ , K(Y ∗ , X ∗ )) ≤ d(T, K(X, Y )) < 1. Thus we only have the possibility, Λ0 ∈ ∂e K(Y ∗ , X ∗ )∗1 . Hence Ruess and Stegall’s theorem ensures that Λ0 = τ ⊗ x∗∗ , as in the proof of Theorem 1. Since in the proof of Theorem 1, we are only manipulating with rank one operators, the conclusion follows. 2 Remark 4. It is clear from the above proof that we only need to assume, extreme points of Y1∗ which attain their norm at a smooth point of Y , have unique norm-preserving extension in Y ∗∗∗ . This assumption is equivalent to the assumption, every smooth point in Y is a very smooth point. Remark 5. We note that if the T in the above theorem is a compact operator and smooth point only in K(X, Y ), then by Theorem 1, we have that T ∗ ∈ K(Y ∗ , X ∗ ) is a smooth point. Since K(Y ∗ , X ∗ ) is an M -ideal in L(Y ∗ , X ∗ ), it is easy to see that T ∗ is a smooth point of L(Y ∗ , X ∗ ). In particular T is also a smooth point of L(X, Y ). It is also easy to see that the same proof goes through, if we assume T ∗ attains its norm and d(T ∗ , K(Y ∗ , X ∗ )) < T . We recall that by Proposition 4 in Section 4 of [7], one has that {T ∈ L(X, Y ) : T ∗ attains its norm} is a dense subset of L(X, Y ).
T.S.S.R.K. Rao / Linear Algebra and its Applications 517 (2017) 129–133
133
References [1] P. Harmand, D. Werner, W. Werner, M -Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993, viii+387 pp. [2] R.B. Holmes, Geometric Functional Analysis and Its Applications, Graduate Texts in Mathematics, vol. 24, Springer-Verlag, New York–Heidelberg, 1975, x+246 pp. [3] K. Paul, D. Sain, P. Ghosh, Birkhoff–James orthogonality and smoothness of bounded linear operators, Linear Algebra Appl. 506 (2016) 551–563. [4] T.S.S.R.K. Rao, Review of ‘Birkhoff–James orthogonality and smoothness of bounded linear operators’, Zbl 06617602, 2016. [5] W. Ruess, C. Stegall, Extreme points in duals of operator spaces, Math. Ann. 261 (1982) 535–546. [6] F. Sullivan, Geometric properties determined by the higher duals of a Banach space, Illinois J. Math. 21 (1977) 315–331. [7] V. Zizler, On some extremal problems in Banach spaces, Math. Scand. 32 (1973) 214–224 (1974).