2021/10/05 by Dantas, Sheldon, Falcó, Javier, Jung, Mingu
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2110.02066
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 α of Schachermayer and β 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-Nikodým property, then X has the G-Bishop-Phelps property for G-invariant operators whenever G ⊆ L(X) is a compact group of isometries on X.