Group Invariant Operators and Some Applications to Norm-Attaining Theory

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