A characterization of a local vector valued Bollob\'as theorem

In this paper, we are interested in giving two characterizations for the so-called property {\bf L}$_{o,o}$, a local vector valued Bollob\'as type theorem. We say that $(X, Y)$ has this property whenever given $\eps>0$ and an operador $T: X \rightarrow Y$, there is $\eta = \eta(\eps, T)$ such that if $x$ satisfies $\|T(x)\|>1 - \eta$, then there exists $x_0 \in S_X$ such that $x_0 \approx x$ and $T$ itself attains its norm at $x_0$. This can be seen as a strong (although local) Bollob\'as theorem for operators. We prove that the pair $(X, Y)$ has the {\bf L}$_{o,o}$ for compact operators if and only if so does $(X, \K)$ for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $(X \pten Y, \K)$ satisfies the {\bf L}$_{o,o}$ for linear functionals under strict convexity or Kadec-Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $(L_p(\mu) \times L_q(\nu); \K)$ cannot satisfy the {\bf L}$_{o,o}$ for bilinear forms.


Introduction
It has now been 60 years since Bishop and Phelps proved that every bounded linear functional can be approximated by norm-attaining ones [2]. Since then, several researchers have been working in norm-attaining theory in many different directions and it is out of doubt one of the most traditional topics in Functional Analysis nowadays. Bollobás [3] pushed further the Bishop-Phelps theorem by proving that if ε > 0, then there exists η(ε) > 0 such that whenever x * ∈ S X * and x ∈ S X satisfy |x * (x)| > 1 − η(ε), then there exist a new functional x * 0 ∈ S X * and a new element x 0 ∈ S X such that (1.1) |x * 0 (x 0 )| = 1, x 0 − x < ε, and x * 0 − x * < ε.
Let us notice the Bishop-Phelps theorem plays an important role in non-reflexive spaces since otherwise every functional attains its norm. On the other hand, Bollobás theorem does make sense in the reflexive setting and, in this case, the functional x * necessarily attains its norm; so it would be natural to wonder whether a version of Bollobás theorem without changing the initial functional x * holds in general, that is, whether it is possible to take x * 0 = x * in (1.1). In a more general situation, we are wondering the following: given ε > 0, is it possible to find η(ε) > 0 such that whenever T ∈ L(X, Y ) with T = 1 and x ∈ S X satisfy T (x) > 1 − η(ε), one can find a new element x 0 ∈ S X such that T (x 0 ) = 1 and x 0 −x < ε? It is easy to see that the pair (X, K) satisfies it whenever X is a uniformly convex Banach space and it turns out that this is in fact a characterization for uniformly convex spaces (see [12,Theorem 2.1]). Nevertheless, there is no way of getting such a similar statement for linear operators: indeed, the authors in [7] proved that if X and Y are real Banach spaces of dimension greater than or equal to 2, then the pair (X, Y ) always fails such a property. Therefore, the only hope for getting positive results in the context of operators would be by considering a weakening of the mentioned property and that was done in [4,8,9,15,16] (and more recently in [5,6] as a tool to get positive results on different norm-attainment notions). More specifically, we have the following property. Definition 1.1. Let X, Y be Banach spaces. We say that the pair (X, Y ) has the L o,o for operators if given ε > 0 and T ∈ L(X, Y ) with T = 1, there exists η(ε, T ) > 0 such that whenever x ∈ S X satisfies T (x) > 1 − η(ε, T ), there exists x 0 ∈ S X such that Notice that if the pair (X, Y ) satisfies such a property, we are saying that every operator has to be norm-attaining and, consequently, the Banach space X must be reflexive by the James theorem. By using a result due to G. Godefroy, V. Montesinos, and V. Zizler [11] and a characterization by C. Franchetti and R. Payá [10], it turns out that the pair (X, K) has the L o,o for linear functionals if and only if X * is strongly subdifferentiable (SSD, for short; see its definition below). On the other hand, at the same way that it happens in the classical norm-attainment theory (see, for instance, [13]), the L o,o was studied for compact operators [15,16]. It is known that whenever X is strictly convex, the L o,o for compact operators is equivalent to saying that the dual X * is Fréchet differentiable (see [16,Theorem 2.3]); and when X satisfies the Kadec-Klee property then (X, Y ) has the L o,o for compact operators for every Banach space Y (see [15,Theorem 2.12]).
Our first aim in the present paper is to generalize [  Our second main result deals with a strengthening of the L o,o in the context of bilinear forms (see [8] x 0 − x < ε, and y 0 − y < ε. It is known that (X × Y ; K) satisfies the L o,o for bilinear forms whenever  [8], respectively). By observing items (a), (b), and (c) above, one might think that the reflexivity of X ⊗ π Y plays an important role here (notice that (c) gives the result for p -spaces exactly when the projective tensor product p ⊗ π q is reflexive (see [14,Corollary 4.24])). And this is indeed not a coincidence: we have the following result, which gives a complete characterization for the L o,o in terms of the reflexivity of X ⊗ π Y and also relates the L o,o in different classes of functions under strict convexity or Kadec-Klee property assumptions on X.
Theorem B. Let X be a strictly convex Banach space or a Banach space satisfying the Kadec-Klee property. Let Y be an arbitrary Banach space and assume that either X or Y enjoys the approximation property. The following are equivalent.
As a consequence of Theorem B, we have that (L p (µ) × L q (ν); K) cannot satisfy the L o,o for bilinear forms for every 1 < p, q < ∞ since L p (µ) ⊗ π L q (ν) is never reflexive (see [14,Theorem 4.21 and Corollary 4.22]). We conclude the paper with a discussion about the relation between the different properties L o,o in B(X × Y, K).

Terminology and Background.
Here will be working with Banach spaces over the real or complex field K. The unit ball and unit sphere of a Banach space X are denoted by B X and S X , respectively. The symbols L(X, Y ) and B(X × Y ; K) stand for the (bounded) linear operators and bilinear forms, respectively. When Y = K, L(X, Y ) becomes simply X * , the topological dual space of X. We say that T ∈ L(X, Y ) attains its norm if T (x 0 ) = T for some x 0 ∈ S X and we say that The norm of X is said to be strongly subdifferentiable (SSD, for short) at the point x ∈ X if the one-side limit lim t→0 + x + th − x t exists uniformly for h ∈ B X . Let us notice that the norm of X is Fréchet differentiable at x if and only it is Gâteux differentiable and SSD at x. When we say that X is SSD we mean that the norm of X is SSD at every x ∈ S X .
The projective tensor product of two Banach spaces X and Y is the completion of X ⊗ Y endowed with the norm given by We denote the projective tensor product of X and Y endowed with the above norm by X ⊗ π Y . It is well-known (and we will be using these facts with no explicit mention throughout the paper) that x ⊗ y = x y for every x ∈ X and y ∈ Y , and that the closed unit ball of X ⊗ π Y is the closed convex hull of the under the action of a bounded bilinear form B as a bounded linear functional on X ⊗ π Y given by and (X ⊗ π Y ) * = L(X, Y * ) under the action of a bounded linear operator T as a bounded linear functional on X ⊗ π Y given by Recall that a Banach space is said to have the approximation property (AP, in short) if for every compact subset K of X and every ε > 0, there exists a finite-rank operator T : X −→ X such that T (x) − x ε for every x ∈ K. We refer the reader to [14] for background on the beautiful tensor products of Banach spaces and approximation properties theories.
Finally, let us recall that a Banach space X satisfies the Kadec-Klee property if weak and norm topologies coincide in the unit sphere of X.

Proofs of Theorems A and B
We start this section by giving the proof of Theorem A.
(ii) ⇒ (i). Suppose that (X, K) has the L o,o for linear functionals. By contradiction, suppose that there exist ε 0 > 0, T ∈ K(X, Y ) with T = 1, and (x n ) ⊆ S X such that n=1 is bounded, we may (and we do) assume that x n w −→ x 0 for some x 0 ∈ B X . Since T is a compact operator, we have that T (x n ) · −→ T (x 0 ). By (2.1), we have that T (x 0 ) = 1 and, in particular, x 0 ∈ S X . Let us take y * 0 ∈ S Y * to be such that y * (T (x 0 )) = T (x 0 ) = 1. Consider x * 0 := T * y * 0 ∈ S X * . Then x * 0 (x 0 ) = T * y * 0 (x 0 ) = y * (T (x 0 )) = 1. Since x n w −→ x 0 , we have that x * 0 (x n ) −→ x * 0 (x 0 ) = 1 as n → ∞. Since (X, K) has the L o,o for linear functionals, there is (x n ) ⊆ S X such that x * 0 (x n ) = 1 and x n − x n → 0 for every n ∈ N. This shows that 1 = x * 0 (x n ) = T * y * 0 (x n ) = y * 0 (T (x n )), that is, 1 = y * 0 (T (x n )) T (x n ) T = 1, that is, T (x n ) = 1 and then x n ∈ NA(B). The convergence x n − x n → 0 yields the desired contradiction. We now present the proof of Theorem B.
For the proofs of (ii) ⇒ (iii) and (iii) ⇒ (i), we assume that both X, Y are real Banach spaces. We invite the reader to go to Remark 2.2 below for some comments on the complex case.
(ii) ⇒ (iii). Suppose that X ⊗ π Y is reflexive and assume that both (X, R) and (Y, R) satisfy the L o,o for linear functionals. By contradiction, let us assume that (X × Y ; R) fails to have the L o,o for bilinear forms. Then, there exist ε 0 > 0, B ∈ B(X × Y ; R) with B = 1, and (x n , y n ) ∞ n=1 ⊆ S X × S Y such that (2.2) 1 B(x n , y n ) 1 − 1 n and whenever (x n , y n ) ⊂ S X × S Y is such that B(x n , y n ) = 1, we have that (2.3) x n − x n ε 0 and y n − y n ε 0 .
Since X and Y are reflexive and both (x n ) ∞ n=1 and (y n ) ∞ n=1 are bounded, we may assume (and we do) that x n w −→ x 0 and y n w −→ y 0 for some x 0 ∈ B X and y 0 ∈ B Y . Now, since X ⊗ π Y is reflexive and we are assuming that either X or Y enjoys the AP, by [14,Theorem 4.21], we have that (X ⊗ π Y ) * = L(X, Y * ) = K(X, Y * ). Let T ∈ K(X, Y * ) be arbitrary. Since x n w −→ x 0 and T is completely continuous (i.e. T (K) is a compact subset of Y * whenever K is a weakly compact subset of X), we have that T (x n ) · −→ T (x 0 ) and then since as n → ∞ for every T ∈ K(X, Y * ) = (X ⊗ π Y ) * . This means that x n ⊗ y n w −→ x 0 ⊗ y 0 . In particular, since B ∈ B(X × Y ; R) = (X ⊗ π Y ) * , we have that B(x n , y n ) −→ B(x 0 , y 0 ) and by (2.2), B(x 0 , y 0 ) = 1. In particular, x 0 ∈ S X and y 0 ∈ S Y .
Let us consider T B ∈ L(X, Y * ) and S B ∈ L(Y, X * ) to be the associated linear operators to the bilinear form B. Therefore, we have that We prove (a) since (b) is analogous. As T * B is a compact operator and y n w −→ y 0 , we Now we prove that x n −x −→ ∞ as n → ∞. Assume first that X satisfies the Kadec-Klee property. Since x 0 ∈ S X and x n w −→ x 0 , we have that x n − x 0 → 0 as n → ∞. We prove that the same holds if X is taken to be strictly convex. Indeed, by using item (a) of Claim, we have that T * B (y 0 )(x n ) −→ 1 as n → ∞. Since (X, R) satisfies the L o,o and X is strictly convex, we have that X * is Fréchet differentiable (see [9,Theorem 2.5.(b)] and then, by theŠmulyan lemma, we have that x n − x 0 −→ 0 as n → ∞ as desired.
Consider the sets and J := I c . Then, This clearly implies that On the other hand, for every n ∈ I, we have that B(x n , y n ) > 1−η(ε, B). Since (X ×Y ; R) has the L o,o for bilinear forms, there is (x n , y n ) ⊆ S X × S Y such that B(x n , y n ) = 1, x n − x n < ε, and y n − y n < ε for every n ∈ I. Let us then define the tensor So, we have that On the other hand, since x n ⊗ y n − x n ⊗ y n x n ⊗ y n − x n ⊗ y n + x n ⊗ y n − x n ⊗ y n y n − y n + x n − x n < 2ε we have, by using (2.5), that z − z π = n∈I λ n x n ⊗ y n − n∈I λ n x n ⊗ y n − n∈J λ n x n ⊗ y n n∈I λ n x n ⊗ y n − x n ⊗ y n + n∈J λ n < 2ε n∈I λ n + n∈J λ n < 2ε(1 + ε 2 ) + ε. Proof. Under the assumption of (a), we have that the projective tensor product p ⊗ π q is reflexive (see [14,Corollary 4.24]). For (b), since L p (µ) ⊗ π L q (ν) contains complemented isomorphic copies of 1 for every p, q, it is never reflexive (see [14,Theorem 4. We have the following relation between properties (A), (B), and (C): • General implications. Clearly, we have that (C) ⇒ (B) by considering the associated bilinear for B T ∈ B(X × Y ; K) of a given operator T ∈ L(X, Y * ). Also, by our Theorem B (implication (iii) ⇒ (i)) and noticing that, for this implication, we do not need any assumption on X besides reflexivity (not even approximation property assumptions, see Remark 2.3), we also have that (C) ⇒ (A).
• Not true implications. (B) does not imply (A) or (C) in general. Indeed, by [1, Theorem 2.4.10], for every 1 < p < ∞, we have that L(c 0 , p ) = K(c 0 , p ) = L( p , 1 ), where p is the conjugate index of p. We have that ( p , 1 ) has the L o,o for operators by Theorem A (since ( p , K) has the L o,o for linear functionals) but neither ( p × c 0 ; K) nor ( p ⊗ π c 0 ; K) can have the L o,o for bilinear forms and for linear functionals, respectively, since c 0 is not reflexive.
• With extra assumptions implications. Assume that either X or Y has the AP. In this case, implication (A) ⇒ (B) holds. Indeed, if (X ⊗ π Y ; K) has the L o,o for linear functionals, then X ⊗ π Y must be reflexive and, by the assumption that X or Y has the AP, every operator from X into Y * is compact and by Theorem B (implication (i) ⇒ (ii)), the pair (X, K) has the L o,o for linear functionals. By Theorem A, the pair (X, Y * ) has the L o,o for operators. Finally, if X or Y has the AP and X or Y is stricly convex, then (A)⇒(C)