Gamma-flatness and Bishop-Phelps-Bollobas type theorems for operators

Abstract

[EN] The Bishop-Phelps-Bollobas property deals with simultaneous approximation of an operator T and a vector x at which T nearly attains its norm by an operator T-o and a vector x(o), respectively, such that T-o attains its norm at x(o). In this note we extend the already known results about the Bishop-Phelps Bollobas property for Asplund operators to a wider class of Banach spaces and to a wider class of operators. Instead of proving a BPB-type theorem for each space separately we isolate two main notions: Gamma-flat operators and Banach spaces with ACK(rho) structure. In particular, we prove a general BPB-type theorem for Gamma-flat operators acting to a space with ACK(rho) structure and show that uniform algebras and spaces with the property beta have ACK(rho) structure. We also study the stability of the ACK(rho) structure under some natural Banach space theory operations. As a consequence, we discover many new examples of spaces Y such that the Bishop-Phelps-Bollobas property for Asplund operators is valid for all pairs of the form (X, Y).