Group invariant operators and some applications on norm-attaining theory

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 $X$.