Norm attaining operators and variational principle

TL;DR

This paper introduces a new linear variational principle and demonstrates that in certain Banach spaces, the set of norm strongly attaining operators is large, complementing a small, porous set, with applications to various operator classes.

Contribution

It extends the variational principle to a broader context and shows the abundance of norm strongly attaining operators in Banach spaces with property (α).

Findings

Set of norm strongly attaining operators is a complement of a $\sigma$-porous set.

Results apply to both linear and nonlinear operator spaces.

The variational principle generalizes previous foundational results.

Abstract

We establish a linear variational principle extending the Deville-Godefroy-Zizler's one. We use this variational principle to prove that if $X$ is a Banach space having property $(\alpha)$ of Schachermayer and $Y$ is any banach space, then the set of all norm strongly attaining linear operators from $X$ into $Y$ is a complement of a $\sigma$-porous set. Moreover, the results of the paper applies also to an abstract class of (linear and nonlinear) operator spaces.