The Bishop–Phelps–Bollobás Property and Absolute Sums

Abstract

In this paper, we study conditions assuring that the Bishop–Phelps–Bollobás 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,(a)if Y 1 is an absolute summand of Y, then (X, Y 1 ) has the BPBp;(b)if X 1 is an absolute summand of X of type 1 or ∞, then (X 1 , 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–Bollobás 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.