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arXiv · 2311.14070

The Bishop--Phelps--Bollob\'as property for weighted holomorphic mappings

A. Jiménez-Vargas·M. I. Ramírez·Moisés Villegas-Vallecillos
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Abstract

Given an open subset $U$ of a complex Banach space $E$, a weight $v$ on $U$ and a complex Banach space $F$, let $H^\infty_v(U,F)$ denote the Banach space of all weighted holomorphic mappings from $U$ into $F$, endowed with the weighted supremum norm. We introduce and study a version of the Bishop--Phelps--Bollob\'as property for $H^\infty_v(U,F)$ ($WH^\infty$-BPB property, for short). A result of Lindenstrauss type with sufficient conditions for $H^\infty_v(U,F)$ to have the $WH^\infty$-BPB property for every space $F$ is stated. This is the case of $H^\infty_{v_p}(\mathbb{D},F)$ with $p\geq 1$, where $v_p$ is the standard polynomial weight on $\mathbb{D}$. The study of the relations of the $WH^\infty$-BPB property for the complex and vector-valued cases is also addressed as well as the extension of the cited property for mappings $f\in H^\infty_v(U,F)$ such that $vf$ has relatively compact range in $F$.

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A. Jiménez-Vargas, M. I. Ramírez, Moisés Villegas-Vallecillos. 2023-11-23. The Bishop--Phelps--Bollob\'as property for weighted holomorphic mappings. https://arxiv.org/abs/2311.14070

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