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 Hv∞(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 Hv∞(U,F) (WH∞-BPB property, for short). A result of Lindenstrauss type with sufficient conditions for Hv∞(U,F) to have the WH∞-BPB property for every space F is stated. This is the case of Hvp∞(D,F) with p≥1, where vp is the standard polynomial weight on D. The study of the relations of the WH∞-BPB property for the complex and vector-valued cases is also addressed as well as the extension of the cited property for mappings f∈Hv∞(U,F) such that vf has a relatively compact range in F.