Weighted holomorphic mappings associated with p-compact type sets

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 f U→ F, under the weighted supremum norm \|f\|ᵥ:=(x)\|f(x)\| x∈ U\. In this paper, we introduce and study the classes of weighted holomorphic mappings H^∞_vKₚ(U,F) (resp., H^∞_vKwp(U,F) and H^∞_vKᵤₚ(U,F)) for which the set $(vf)(U)$ is relatively p-compact (resp., relatively weakly p-compact and relatively unconditionally p-compact). We prove that these mapping classes are characterized by p-compact (resp., weakly p-compact and unconditionally p-compact) linear operators defined on a Banach predual space of H^∞ᵥ(U) by linearization. We show that H^∞_vKₚ (resp., H^∞_vKwp and H^∞_vKᵤₚ) is a Banach ideal of weighted holomorphic mappings which is generated by composition with the ideal of p-compact (resp., weakly p-compact and unconditionally p-compact) linear operators and contains the Banach ideal of all right p-nuclear weighted holomorphic mappings. We also prove that these weighted holomorphic mappings can be factorized through a quotient space of lp^*, and f^∞_vKₚ(U,F) (resp., f^∞_vKᵤₚ(U,F)) if and only if its transposition fᵗ is quasi p-nuclear (resp., quasi unconditionally p-nuclear).