The Bishop-Phelps-Bollob\'as property and absolute sums

TL;DR

This paper investigates how the Bishop-Phelps-Bollobás property (BPBp) and related properties are preserved when considering absolute summands of Banach spaces, extending known inheritance results to various operator classes.

Contribution

It establishes new inheritance results for the BPBp and related properties for absolute summands of Banach spaces, including for compact operators and numerical radius, expanding the understanding of these properties.

Findings

BPBp is inherited by absolute summands of the range or domain space.

Results extend to the BPBp for compact operators and norm attaining operators.

Inheritance also holds for the numerical radius and related properties.

Abstract

In this paper we study conditions assuring that the Bishop-Phelps-Bollob\'as 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 Y1 is an absolute summand of Y, then (X, Y1) has the BPBp; (b) if X1 is an absolute summand of X of type 1 or \infty, 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-Bollob\'as property for numerical radius is inherited by absolute summands of type 1 or \infty. Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.