Weighted holomorphic mappings associated with p-compact type sets

Abstract

Given an open subset $U$ of a complex Banach space $E$, a weight $v$ on $U$, and a complex Banach space $F$, let $\mathcal{H}^\infty_v(U,F)$ denote the Banach space of all weighted holomorphic mappings $f\colon U\to F$, under the weighted supremum norm $\left\|f\right\|_v:=\sup\left\{v(x)\left\|f(x)\right\|\colon x\in U\right\}$. In this paper, we introduce and study the classes of weighted holomorphic mappings $\mathcal{H}^\infty_{v\mathcal{K}_{p}}(U,F)$ (resp., $\mathcal{H}^\infty_{v\mathcal{K}_{wp}}(U,F)$ and $\mathcal{H}^\infty_{v\mathcal{K}_{up}}(U,F)$) 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 $\mathcal{H}^\infty_v(U)$ by linearization. We show that $\mathcal{H}^\infty_{v\mathcal{K}_{p}}$ (resp., $\mathcal{H}^\infty_{v\mathcal{K}_{wp}}$ and $\mathcal{H}^\infty_{v\mathcal{K}_{up}}$) 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 $l_{p^*}$, and $f\in\mathcal{H}^\infty_{v\mathcal{K}_{p}}(U,F)$ (resp., $f\in\mathcal{H}^\infty_{v\mathcal{K}_{up}}(U,F))$ if and only if its transposition $f^t$ is quasi $p$-nuclear (resp., quasi unconditionally $p$-nuclear).