Bollobás-type theorems for range strongly exposing operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912757577678848 |
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| author | Del Río, Helena |
| author_facet | Del Río, Helena |
| contents | We study Bollobás-type theorems for range strongly exposing operators. When such a theorem holds for operators from a Banach space $X$ into another Banach space $Y$, we say that the pair $(X,Y)$ satisfies the Bishop-Phelps-Bollobás property for range strongly exposing operators (BPBp-RSE, for short). We provide new characterisations of uniform convexity and complex uniform convexity via the BPBp-RSE, including for pairs involving spaces such as $L_1(μ), L_\infty(μ)$ and $c_0$. In particular, we show that $(L_1(μ), Y)$ satisfies the BPBp-RSE if and only if $Y$ is uniformly convex, and that $(L_\infty(μ), Y)$ or $(c_0, Y)$ satisfy the BPBp-RSE if and only if $Y$ is $\mathbb{C}$-uniformly convex. We also highlight differences between the real and complex cases, showing that there exist pairs $(X, Y)$ for which the BPBp-RSE holds in the complex setting but fails for their respective underlying real spaces. Additionally, we consider various subspaces of operators, such as compact and finite-rank, and extend several results from the literature to this new setting. The paper concludes with a collection of open problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bollobás-type theorems for range strongly exposing operators Del Río, Helena Functional Analysis 46B04 (Primary) 46B20, 46B22, 46B25, 47B01, 47B07 (Secondary) We study Bollobás-type theorems for range strongly exposing operators. When such a theorem holds for operators from a Banach space $X$ into another Banach space $Y$, we say that the pair $(X,Y)$ satisfies the Bishop-Phelps-Bollobás property for range strongly exposing operators (BPBp-RSE, for short). We provide new characterisations of uniform convexity and complex uniform convexity via the BPBp-RSE, including for pairs involving spaces such as $L_1(μ), L_\infty(μ)$ and $c_0$. In particular, we show that $(L_1(μ), Y)$ satisfies the BPBp-RSE if and only if $Y$ is uniformly convex, and that $(L_\infty(μ), Y)$ or $(c_0, Y)$ satisfy the BPBp-RSE if and only if $Y$ is $\mathbb{C}$-uniformly convex. We also highlight differences between the real and complex cases, showing that there exist pairs $(X, Y)$ for which the BPBp-RSE holds in the complex setting but fails for their respective underlying real spaces. Additionally, we consider various subspaces of operators, such as compact and finite-rank, and extend several results from the literature to this new setting. The paper concludes with a collection of open problems. |
| title | Bollobás-type theorems for range strongly exposing operators |
| topic | Functional Analysis 46B04 (Primary) 46B20, 46B22, 46B25, 47B01, 47B07 (Secondary) |
| url | https://arxiv.org/abs/2512.10442 |