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