Geometry of the space of compact operators endowed with the numerical radius norm
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912235523145728 |
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| author | Han, Manwook Kim, Sun Kwang |
| author_facet | Han, Manwook Kim, Sun Kwang |
| contents | We investigate the space of bounded linear operators on a Banach space equipped with a norm which is equivalent to the operator norm such that the subspace of compact operators is an M-ideal. In particular, we observe that the space of compact operators on $\ell_p$ $(1<p<\infty)$ equipped with the numerical radius norm is an M-ideal whenever the numerical index of $\ell_p$ is not $0$. On the other hand, we show that the space of compact operators on a Banach space containing an isomorphic copy of $\ell_1$ whose numerical index is greater than $1/2$ is not M-ideals. We also study the proximinality, the existence of farthest points and the compact perturbation property for the numerical radius. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometry of the space of compact operators endowed with the numerical radius norm Han, Manwook Kim, Sun Kwang Functional Analysis 46Bxx We investigate the space of bounded linear operators on a Banach space equipped with a norm which is equivalent to the operator norm such that the subspace of compact operators is an M-ideal. In particular, we observe that the space of compact operators on $\ell_p$ $(1<p<\infty)$ equipped with the numerical radius norm is an M-ideal whenever the numerical index of $\ell_p$ is not $0$. On the other hand, we show that the space of compact operators on a Banach space containing an isomorphic copy of $\ell_1$ whose numerical index is greater than $1/2$ is not M-ideals. We also study the proximinality, the existence of farthest points and the compact perturbation property for the numerical radius. |
| title | Geometry of the space of compact operators endowed with the numerical radius norm |
| topic | Functional Analysis 46Bxx |
| url | https://arxiv.org/abs/2502.10821 |