Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP
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arXiv
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| Natura: | Preprint |
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2025
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| author | Grando, Thiago Santos, Elisa R. |
| author_facet | Grando, Thiago Santos, Elisa R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17762 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP Grando, Thiago Santos, Elisa R. Functional Analysis Primary 46B20, secondary 46B25, 46E40 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. |
| title | Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP |
| topic | Functional Analysis Primary 46B20, secondary 46B25, 46E40 |
| url | https://arxiv.org/abs/2506.17762 |