The Bishop–Phelps–Bollobás Property for Weighted Holomorphic Mappings

期刊论文
出版模式：

开放获取
发表日期：

2024年5月11日

查看全文

A. Jiménez-Vargas,
M. I. Ramírez,
Moisés Villegas-Vallecillos

摘要

Given an open subset U of a complex Banach space E, a weight v on U and a complex Banach space F, let $H^\infty _v(U,F)$ $H_{v}^{\infty} (U, F)$ 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–Bollobás property for $H^\infty _v(U,F)$ $H_{v}^{\infty} (U, F)$ ( $WH^\infty $ $W H^{\infty}$ -BPB property, for short). A result of Lindenstrauss type with sufficient conditions for $H^\infty _v(U,F)$ $H_{v}^{\infty} (U, F)$ to have the $WH^\infty $ $W H^{\infty}$ -BPB property for every space F is stated. This is the case of $H^\infty _{v_p}(\mathbb {D},F)$ $H_{v_{p}}^{\infty} (D, F)$ with $p\ge 1$ $p \geq 1$ , where $v_p$ $v_{p}$ is the standard polynomial weight on $\mathbb {D}$ $D$ . The study of the relations of the $WH^\infty $ $W H^{\infty}$ -BPB property for the complex and vector-valued cases is also addressed as well as the extension of the cited property for mappings $f\in H^\infty _v(U,F)$ $f \in H_{v}^{\infty} (U, F)$ such that vf has a relatively compact range in F.