On a set of norm attaining operators and the strong Birkhoff-James orthogonality
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| Format: | Preprint |
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2022
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| _version_ | 1866910884461281280 |
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| author | Choi, Geunsu Jung, Mingu Kim, Sun Kwang |
| author_facet | Choi, Geunsu Jung, Mingu Kim, Sun Kwang |
| contents | Continuing the study of recent results on the Birkhoff-James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia-Šemrl property for operators which is weaker than the Bhatia-Šemrl property. The set of operators with the adjusted Bhatia-Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia-Šemrl property. It is known that the set of operators with the Bhatia-Šemrl property is norm-dense if the domain space $X$ of the operators has the Radon-Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $c_0$, $L_1[0,1]$ and $C[0,1]$. In contrast with the Bhatia-Šemrl property, we show that the set of operators with the adjusted Bhatia-Šemrl property is norm-dense when the domain space is $c_0$ or $L_1[0,1]$. Moreover, we show that the set of functionals having the adjusted Bhatia-Šemrl property on $C[0,1]$ is not norm-dense but such a set is weak-$*$-dense in $C(K)^*$ for any compact Hausdorff $K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_11987 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On a set of norm attaining operators and the strong Birkhoff-James orthogonality Choi, Geunsu Jung, Mingu Kim, Sun Kwang Functional Analysis Primary: 46B04, Secondary: 46B20, 46B25 Continuing the study of recent results on the Birkhoff-James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia-Šemrl property for operators which is weaker than the Bhatia-Šemrl property. The set of operators with the adjusted Bhatia-Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia-Šemrl property. It is known that the set of operators with the Bhatia-Šemrl property is norm-dense if the domain space $X$ of the operators has the Radon-Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $c_0$, $L_1[0,1]$ and $C[0,1]$. In contrast with the Bhatia-Šemrl property, we show that the set of operators with the adjusted Bhatia-Šemrl property is norm-dense when the domain space is $c_0$ or $L_1[0,1]$. Moreover, we show that the set of functionals having the adjusted Bhatia-Šemrl property on $C[0,1]$ is not norm-dense but such a set is weak-$*$-dense in $C(K)^*$ for any compact Hausdorff $K$. |
| title | On a set of norm attaining operators and the strong Birkhoff-James orthogonality |
| topic | Functional Analysis Primary: 46B04, Secondary: 46B20, 46B25 |
| url | https://arxiv.org/abs/2208.11987 |