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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.01302 |
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| _version_ | 1866909278782095360 |
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| author | García, Domingo Maestre, Manuel Martín, Miguel Roldán, Óscar . |
| author_facet | García, Domingo Maestre, Manuel Martín, Miguel Roldán, Óscar . |
| contents | We study the set $\operatorname{MA}(X,Y)$ of operators between Banach spaces $X$ and $Y$ that attain their minimum norm, and the set $\operatorname{QMA}(X,Y)$ of operators that quasi attain their minimum norm. We characterize the Radon-Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $\operatorname{MA}(X,Y)$ and $\operatorname{QMA}(X,Y)$. We show that every infinite-dimensional Banach space $X$ has an isomorphic space $Y$ such that not every operator from $X$ to $Y$ quasi attains its minimum norm. We introduce and study Bishop-Phelps-Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01302 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On density and Bishop-Phelps-Bollobás type properties for the minimum norm García, Domingo Maestre, Manuel Martín, Miguel Roldán, Óscar . Functional Analysis 46B04 (primary), 46B03, 46B20, 46B22, 46B25 (secondary) We study the set $\operatorname{MA}(X,Y)$ of operators between Banach spaces $X$ and $Y$ that attain their minimum norm, and the set $\operatorname{QMA}(X,Y)$ of operators that quasi attain their minimum norm. We characterize the Radon-Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $\operatorname{MA}(X,Y)$ and $\operatorname{QMA}(X,Y)$. We show that every infinite-dimensional Banach space $X$ has an isomorphic space $Y$ such that not every operator from $X$ to $Y$ quasi attains its minimum norm. We introduce and study Bishop-Phelps-Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them. |
| title | On density and Bishop-Phelps-Bollobás type properties for the minimum norm |
| topic | Functional Analysis 46B04 (primary), 46B03, 46B20, 46B22, 46B25 (secondary) |
| url | https://arxiv.org/abs/2405.01302 |