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 functionals and operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorem and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators, establishing group invariant versions of the properties α of Schachermayer and β of Lindenstrauss, and presenting relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain's result, establishing that if X has the Radon-Nikodým property, then X has the G-Bishop-Phelps theorem for G-invariant operators whenever G is a compact group.