Stability results of the Bishop-Phelps-Bollob\'as property and the generalized AHSP

TL;DR

This paper explores the stability of the Bishop-Phelps-Bollobás property for operators in Banach spaces, introducing the generalized AHSP and establishing conditions under which the property is preserved across various function space pairs.

Contribution

It introduces the generalized AHSP for Banach space pairs and proves stability results of the BPBp under certain space constructions and conditions.

Findings

BPBp stability for $(X, ext{C}_0(L,Y))$ implies stability for $(X,Y)$.

Generalized AHSP stability under isometric operator conditions.

Extension of stability results to spaces like $ ext{C}(K,Y)$, $ ext{C}_0(L,Y)$, and $ ext{C}_b( ext{Omega},Y)$.

Abstract

In this paper, we study the Bishop-Phelps-Bollob\'as property for operators (BPBp for short). To this end, we investigate the generalized approximate hyperplane series property (generalized AHSP for short) for a pair $(X,Y)$ of Banach spaces, which characterizes when $(\ell_1(X),Y)$ has the BPBp. We prove the following results. For a locally compact Hausdorff space $L$, if $(X, \mathcal{C}_0(L,Y))$ has the BPBp, then so does $(X,Y)$. Furthermore, if the pair $(X, Y)$ has the generalized AHSP and $\mathcal{L}(X,Z) = \mathcal{K}(X,Z)$, then the pair $(X, Z)$ also has the generalized AHSP, where $Z$ is one of the spaces $\mathcal{C}(K, Y)$, $\mathcal{C}_0(L, Y)$, or $\mathcal{C}_b(\Omega, Y)$, with $K$ a compact Hausdorff space and $\Omega$ a completely regular space.