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

Thiago Grando; Elisa R. Santos

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arXiv · 2506.17762

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

Thiago Grando·Elisa R. Santos

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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.

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Thiago Grando, Elisa R. Santos. 2025-06-21. Stability results of the Bishop-Phelps-Bollob\'as property and the generalized AHSP. https://arxiv.org/abs/2506.17762

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