On the existence of non-norm-attaining operators

TL;DR

This paper establishes conditions for the existence of operators that do not attain their norm in Banach spaces, generalizing previous results and linking norm-attainment to topological properties and the Schur property.

Contribution

It provides necessary and sufficient conditions for non-norm-attaining operators using Pfitzner's theorem and generalizes earlier findings on norm-attainment in Banach spaces.

Findings

Characterization of non-norm-attaining operators via weak operator topology

Equivalence of conditions for norm-attainment when the pair has the bounded compact approximation property

Characterization of the Schur property through norm-attaining operators

Abstract

In this paper we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal{L}(E, F)$. By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set $K$ of $\mathcal{L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm closed convex hull, then we can guarantee the existence of an operator which does not attain its norm. This allows us to provide the following generalization of results due to Holub and Mujica. If $E$ is a reflexive space, $F$ is an arbitrary Banach space, and the pair $(E, F)$ has the bounded compact approximation property, then the following are equivalent: (i) $\mathcal{K}(E, F) = \mathcal{L}(E, F)$; (ii) Every operator from $E$ into $F$ attains its norm; (iii) $(\mathcal{L}(E,F), \tau_c)^* = (\mathcal{L}(E, F), \| \cdot \|)^*$; where $\tau_c$ denotes the topology of compact convergence. We conclude the paper presenting a characterization of the Schur property in terms of norm-attaining operators.