Group invariant operators and some applications on norm-attaining theory

TL;DR

This paper explores the geometric structure of group invariant operators between Banach spaces, extending classical theorems and properties to this setting, and applying these to norm-attaining operator theory.

Contribution

It introduces group invariant versions of key theorems and properties, broadening the scope of norm-attaining operator results to include group invariance.

Findings

Established group invariant Hahn-Banach separation theorems

Generalized properties of Schachermayer and Lindenstrauss to the invariant setting

Extended Bourgain's result on Radon-Nikodým property and G-Bishop-Phelps property

Abstract

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties $\alpha$ of Schachermayer and $\beta$ of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain's result, which says that if $X$ has the Radon-Nikod\'ym property, then $X$ has the $G$-Bishop-Phelps property for $G$-invariant operators whenever $G \subseteq \mathcal{L}(X)$ is a compact group of isometries on $X$.