The Bishop–Phelps–Bollobás Property for Weighted Holomorphic Mappings

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) 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–Bollobás property for H^∞ ᵥ(U,F) H v ∞ ( U , F ) ( WH^∞ W H ∞ -BPB property, for short). A result of Lindenstrauss type with sufficient conditions for H^∞ ᵥ(U,F) H v ∞ ( U , F ) to have the WH^∞ W H ∞ -BPB property for every space F is stated. This is the case of H^∞ vₚ(D,F) H v p ∞ ( D , F ) with p≥ 1 p ≥ 1 , where vₚ v p is the standard polynomial weight on D D . The study of the relations of the WH^∞ W H ∞ -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) f ∈ H v ∞ ( U , F ) such that vf has a relatively compact range in F .