The Bishop–Phelps–Bollobás Property for Weighted Holomorphic Mappings

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–Bollobás 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 \(p\ge 1\), 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 a relatively compact range in F.