A characterization of Banach spaces with numerical index one
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908936813150208 |
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| author | Pal, Subhadip Roy, Saikat Sain, Debmalya |
| author_facet | Pal, Subhadip Roy, Saikat Sain, Debmalya |
| contents | We investigate the extremal properties of the unit ball of $L(X)_w^*$, the dual space of bounded linear operators defined on a Banach space $X$ equipped with the numerical radius norm. As an application of the present study, we obtain a geometric characterization of Banach spaces with numerical index one, which extends the well-known McGregor's characterization of finite-dimensional Banach spaces with numerical index one. We also present refinements of several earlier results in this direction, including an explicit description of the extreme points of $B_{L(X)_w^*}$, the unit ball of $L(X)_w^*$, for any finite-dimensional Banach space $X$. This allows us to obtain an independent and elementary proof of McGregor's characterization of finite-dimensional Banach spaces with numerical index one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A characterization of Banach spaces with numerical index one Pal, Subhadip Roy, Saikat Sain, Debmalya Functional Analysis 46B20, 47A12, 47B01 We investigate the extremal properties of the unit ball of $L(X)_w^*$, the dual space of bounded linear operators defined on a Banach space $X$ equipped with the numerical radius norm. As an application of the present study, we obtain a geometric characterization of Banach spaces with numerical index one, which extends the well-known McGregor's characterization of finite-dimensional Banach spaces with numerical index one. We also present refinements of several earlier results in this direction, including an explicit description of the extreme points of $B_{L(X)_w^*}$, the unit ball of $L(X)_w^*$, for any finite-dimensional Banach space $X$. This allows us to obtain an independent and elementary proof of McGregor's characterization of finite-dimensional Banach spaces with numerical index one. |
| title | A characterization of Banach spaces with numerical index one |
| topic | Functional Analysis 46B20, 47A12, 47B01 |
| url | https://arxiv.org/abs/2604.03690 |