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^∞_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^∞_v (U, F) (WH^∞-BPB property, for short). A result of Lindenstrauss type with sufficient conditions for H^∞_v (U, F) to have the WH^∞-BPB property for every space F is stated. This is the case of H^∞_ {v_p} (D, F) with p ≥ 1, where v_p 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 ∈ H^∞_v (U, F) such that vfhas a relatively compact range in F.