The Bishop-Phelps-Bollobas Property and Absolute Sums

Abstract

In this paper, we study conditions assuring that the Bishop-Phelps-Bollobas property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X,Y) of Banach spaces having the BPBp, if Y1 is an absolute summand of Y, then (X,Y1) has the BPBp;if X1 is an absolute summand of X of type 1 or , then (X1,Y) has the BPBp. Besides, analogous results for the BPBp for compact operators and for the density of norm-attaining operators are also given. We also show that the Bishop-Phelps-Bollobas property for numerical radius is inherited by absolute summands of type 1 or . Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.