Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP

TL;DR

The paper investigates the stability of the Bishop-Phelps-Bollobás property for operators (BPBp) and the generalized approximate hyperplane series property (generalized AHSP) for pairs of Banach spaces. It proves that for locally compact $L$, the BPBp passes from $(X,\mathcal{C}_0(L,Y))$ to $(X,Y)$, but the converse may fail, and that under $\mathcal{L}(X,Z)=\mathcal{K}(X,Z)$ the generalized AHSP transfers from $(X,Y)$ to $(X,Z)$ when $Z$ is in $\mathcal{C}(K,Y)$, $\mathcal{C}_0(L,Y)$, or $\mathcal{C}_b(\Omega,Y)$ (with $K$ compact Hausdorff and $\Omega$ completely regular). These results extend prior work and provide transfer principles linking operator BPBp and generalized AHSP with spaces of continuous functions. The methods draw on techniques from ACKLM15 and CGKM14, broadening the understanding of norm-attaining operators in function-space settings.

Abstract

In this paper, we study the Bishop-Phelps-Bollobás 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 $\mathcal{C}_b(Ω, Y)$, with $K$ a compact Hausdorff space and $Ω$ a completely regular space.

Theorems & Definitions (13)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.3
• Remark 1.4
• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof