On composition ideals and dual ideals of bounded holomorphic mappings

TL;DR

This paper explores the structure of bounded holomorphic mappings via composition and dual ideals, introducing new classes like p-integral and p-nuclear mappings, and analyzing their properties such as compactness.

Contribution

It introduces the bounded-holomorphic dual ideal, characterizes its elements, and studies new holomorphic mapping ideals with properties like weak compactness and compactness.

Findings

Characterization of elements in the bounded-holomorphic dual ideal.

Every p-integral holomorphic mapping has relatively weakly compact range.

Every p-nuclear holomorphic mapping has compact range.

Abstract

Applying a linearization theorem due to J. Mujica, we study the ideals of bounded holomorphic mappings $\mathcal{H}^\infty\circ\mathcal{I}$ generated by composition with an operator ideal $\mathcal{I}$. The bounded-holomorphic dual ideal of $\mathcal{I}$ is introduced and its elements are characterized as those that admit a factorization through $\mathcal{I}^\mathrm{dual}$. For complex Banach spaces $E$ and $F$, we also analyze new ideals of bounded holomorphic mappings from an open subset $U\subseteq E$ to $F$ such as $p$-integral holomorphic mappings and $p$-nuclear holomorphic mappings with $1\leq p<\infty$. We prove that every $p$-integral ($p$-nuclear) holomorphic mapping from $U$ to $F$ has relatively weakly compact (compact) range.