On the Existence of Non-Norm-Attaining Operators

2003

The Banach space $E$ has the weakly compact approximation property (W.A.P. for short) if there is a constant $C < \infty$ so that for any weakly compact set $D \subset E$ and $\epsilon > 0$ there is a weakly compact operator $V: E \to E$ satisfying $\sup_{x\in D} || x - Vx || < \epsilon$ and $|| V|| \leq C$. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space $J$) have the W.A.P, but that James' tree space $JT$ fails to have the W.A.P. It is also shown that the dual $J^*$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^*)$, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space $Y$ so that $Y$ fails to have the W.A.P., but $Y$ has this approximation property without the uniform bound $C$.

Journal of Mathematical Analysis and Applications, 1998

If a strictly convex Banach space Y contains either a symmetric basic sequence which is not equivalent to the l -basis or a normalized sequence with an upper 1 p-estimate, then there is a Banach space X such that the set of norm-attaining operators is not dense in the Banach space of all bounded linear operators from X into Y. We deduce that no infinite-dimensional uniformly convex Banach space has Lindenstrauss' property B. ᮊ

2016

We prove that the kernel of a quotient operator from an L 1-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case ℓ 1-and Figiel, Johnson and Pe lczyński-case X * separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1-space onto X has the BAP then every kernel of every quotient map from any L 1-space onto X has the BAP. The dual result for L∞-spaces also hold: if for some L∞-space E some quotient E/X has the BAP then for every L∞-space E every quotient E/X has the BAP.

Illinois Journal of Mathematics

We develop a unified approach to characterize approximation properties defined by spaces of operators. Our main result describes them in terms of the approximability of weak*weak continuous operators. In particular, we prove that if A and B are operator ideals satisfying A•B * ⊂ K, then the A(X, X)approximation property of a Banach space X is equivalent to the following "metric" condition: for every Banach space Y and for every operator T ∈ B * (Y, X), there exists a net (Sα) ⊂ A(X, X) such that sup α SαT ≤ T and T * S * α → T * in the strong operator topology on L(X * , Y * ). As application, approximation properties of dual spaces and weak metric approximation properties are studied.

Israel Journal of Mathematics, 2013

We prove that the kernel of a quotient operator from an L 1 -space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky -case ℓ 1 -and Figiel, Johnson and Pe lczyński -case X * separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1 -space onto X has the BAP then every kernel of every quotient map from any L 1 -space onto X has the BAP. The dual result for L∞-spaces also hold: if for some L∞-space E some quotient E/X has the BAP then for every L∞-space E every quotient E/X has the BAP.

Pure and Applied Mathematics, 2018

We have introduced two new notions of convexity and closedness in functional analysis. Let X be a real normed space, then C(⊆ X) is functionally convex (briefly, F-convex), if T (C) ⊆ R is convex for all bounded linear transformations T ∈ B(X, R); and K(⊆ X) is functionally closed (briefly, F-closed), if T (K) ⊆ R is closed for all bounded linear transformations T ∈ B(X, R). By using these new notions, the Alaoglu-Bourbaki-Eberlein-Šmuljan theorem has been generalized. Moreover, we show that X is reflexive if and only if the closed unit ball of X is F-closed. James showed that for every closed convex subset C of a Banach space X, C is weakly compact if and only if every f ∈ X * attains its supremum over C at some point of C. Now, we show that if A is an F-convex subset of a Banach space X, then A is bounded and F-closed if and only if every element of X * attains its supremum over A at some point of A.

2002

1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn 1/p−1/q-complemented in X (cf. [1]). Szankowski [2] has showed that if T(X) = sup{p: X of type p} ̸ = 2 or C(X) = inf{q: X of cotype q} ̸ = 2, then X has a subspace without the approximation property. Thus, if each subspace of X possesses the approximation property, then necessarily T(X) = C(X) = 2 and, therefore, all of finite dimensional subspaces in X are ”well ” complemented. Moreover, the space X need not be Hilbertian (or isomorphic to a Hilbert space). The examples of such spaces were constructed by Johnson [3]. In connection with the examples of Szankowski and Johnson, the natural questions arises: 1)if T(X) = C(X) = 2, then is it true that every subspace in X has the approximation property? 2) how ”well complemented ” may be the finite dimensional subspaces of a space without the approximation property: in particular, does there exist a space X without the approximation property...

1999

Let $Z$ be a fixed separable operator space, $X\subset Y$ general separable operator spaces, and $T:X\to Z$ a completely bounded map. $Z$ is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to $Y$; the Mixed Separable Extension Property (MSEP) if every such $T$ admits a bounded extension to $Y$. Finally, $Z$ is said to have the Complete Separable Complementation Property (CSCP) if $Z$ is locally reflexive and $T$ admits a completely bounded extension to $Y$ provided $Y$ is locally reflexive and $T$ is a complete surjective isomorphism. Let ${\bf K}$ denote the space of compact operators on separable Hilbert space and ${\bf K}_0$ the $c_0$ sum of ${\Cal M}_n$'s (the space of ``small compact operators''). It is proved that ${\bf K}$ has the CSCP, using the second author's previous result that ${\bf K}_0$ has this property. A new proof is given for the result (due to E. Kirchberg) that ${\bf K}_0$ (and hence ${\bf K}$) fails the CSEP. It remains an open question if ${\bf K}$ has the MSEP; it is proved this is equivalent to whether ${\bf K}_0$ has this property. A new Banach space concept, Extendable Local Reflexivity (ELR), is introduced to study this problem. Further complements and open problems are discussed.

Annals of Functional Analysis, 2014

Mathematische Annalen, 2004

Suppose ∞ → X. We construct examples of bounded sets M ⊂ X, such that M w * ⊂ X + 1 2 B X * , but coM w * ⊂ X + αB X * * for any α < 1. These examples show that the previous results of the authors on quantitative versions of Krein's theorem are optimal.