1. On density and Bishop-Phelps-Bollob\'as type properties for the minimum norm
- Author
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García, Domingo, Maestre, Manuel, Martín, Miguel, and Roldán, Óscar
- Subjects
Mathematics - Functional Analysis ,46B04 (primary), 46B03, 46B20, 46B22, 46B25 (secondary) - Abstract
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-Bollob\'as 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., Comment: 22 pages including references. 1 figure
- Published
- 2024
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