On p-summability in weighted Banach spaces of holomorphic functions

TL;DR

This paper introduces and analyzes the class of p-summing weighted holomorphic mappings in Banach spaces, establishing their properties and duality relations, and extending classical theorems to this setting.

Contribution

It defines p-summing weighted holomorphic mappings, proves they form an injective Banach ideal, and extends key theorems like Pietsch Domination and Maurey Extrapolation to this context.

Findings

p-summing weighted holomorphic mappings form an injective Banach ideal.

Variants of Pietsch Domination, Factorization, and Maurey Extrapolation are established.

Duals of p-summing mappings are characterized via weighted holomorphic molecules and tensor norms.

Abstract

Given an open subset $U$ of a complex Banach space $E$, a weight $v$ on $U$, and a complex Banach space $F$, let $\Hv(U,F)$ 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 class $\Pi_p^{\Hv}(U,F)$ of $p$-summing weighted holomorphic mappings. We prove that it is an injective Banach ideal of weighted holomorphic mappings. Variants for weighted holomorphic mappings of Pietsch Domination Theorem, Pietsch Factorization Theorem and Maurey Extrapolation Theorem are presented. We also identify the spaces of $p$-summing weighted holomorphic mappings from $U$ into $F^*$ under the norm $\pi^{\Hv}_p$ with the duals of $F$-valued $\Hv$-molecules on $U$ under a suitable version $d^{\Hv}_{p^*}$ of the Chevet--Saphar tensor norms.