The Bishop-Phelps-Bollobás property for the numerical radius: a Zizler-type approach
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| Format: | Preprint |
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2026
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| _version_ | 1866908946099339264 |
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| author | Kim, Sun Kwang Lee, Han Ju Martin, Miguel Roldan, Oscar |
| author_facet | Kim, Sun Kwang Lee, Han Ju Martin, Miguel Roldan, Oscar |
| contents | We investigate the Bishop-Phelps-Bollobás property for the numerical radius (BPBp-nu) through a Zizler-type perspective on the classical Bishop-Phelps-Bollobás property (BPBp). This approach allows us to establish two new results: the real Banach space $\ell_\infty$ satisfies the BPBp-nu, while the complex space $\ell_1 \oplus_\infty c_0$ does not. Note that the latter provides the first natural example (constructed without renorming techniques) of a Banach space where the numerical radius attaining operators are dense but the BPBp-nu fails. Along the way, we strengthen the main results of the paper [Kim et al, On the Bishop-Phelps-Bollobás theorem for operators and numerical radius, Studia Math., 2016] concerning the interplay between the BPBp for the pair $(X,Y)$ and the BPBp-nu for a direct sum $X\oplus Y$ of Banach spaces. We further explore the validity of the Zizler-type BPBp across different pairs of Banach spaces, and how this property relates to the classical BPBp and the BPBp-nu. Finally, we specialize our analysis to the framework of compact operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_06625 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Bishop-Phelps-Bollobás property for the numerical radius: a Zizler-type approach Kim, Sun Kwang Lee, Han Ju Martin, Miguel Roldan, Oscar Functional Analysis Primary 46B20, Secondary 46B04, 46B22 We investigate the Bishop-Phelps-Bollobás property for the numerical radius (BPBp-nu) through a Zizler-type perspective on the classical Bishop-Phelps-Bollobás property (BPBp). This approach allows us to establish two new results: the real Banach space $\ell_\infty$ satisfies the BPBp-nu, while the complex space $\ell_1 \oplus_\infty c_0$ does not. Note that the latter provides the first natural example (constructed without renorming techniques) of a Banach space where the numerical radius attaining operators are dense but the BPBp-nu fails. Along the way, we strengthen the main results of the paper [Kim et al, On the Bishop-Phelps-Bollobás theorem for operators and numerical radius, Studia Math., 2016] concerning the interplay between the BPBp for the pair $(X,Y)$ and the BPBp-nu for a direct sum $X\oplus Y$ of Banach spaces. We further explore the validity of the Zizler-type BPBp across different pairs of Banach spaces, and how this property relates to the classical BPBp and the BPBp-nu. Finally, we specialize our analysis to the framework of compact operators. |
| title | The Bishop-Phelps-Bollobás property for the numerical radius: a Zizler-type approach |
| topic | Functional Analysis Primary 46B20, Secondary 46B04, 46B22 |
| url | https://arxiv.org/abs/2604.06625 |