On holomorphic mappings with relatively $p$-compact range

TL;DR

This paper characterizes holomorphic mappings with relatively p-compact ranges in Banach spaces, showing they form a Banach ideal generated by p-compact operators and relate to p-nuclear mappings.

Contribution

It introduces a complete characterization of holomorphic mappings with relatively p-compact ranges and establishes their structure as a Banach ideal generated by p-compact operators.

Findings

Holomorphic mappings with relatively p-compact ranges are characterized via Mujica's linearisations.

The class forms a surjective Banach ideal generated by composition with p-compact operators.

Mappings with relatively p-compact ranges factor through quotients of ℓ_{p^*} or have quasi p-nuclear transposes.

Abstract

Related to the concept of $p$-compact operator with $p\in [1,\infty]$ introduced by Sinha and Karn, this paper deals with the space $\mathcal{H}^\infty_{\mathcal{K}_p}(U,F)$ of all Banach-valued holomorphic mappings on an open subset $U$ of a complex Banach space $E$ whose ranges are relatively $p$-compact subsets of $F$. We characterize such holomorphic mappings as those whose Mujica's linearisations on the canonical predual of $\mathcal{H}^\infty(U)$ are $p$-compact operators. This fact allows us to make a complete study of them. We show that $\mathcal{H}^\infty_{\mathcal{K}_p}$ is a surjective Banach ideal of bounded holomorphic mappings which is generated by composition with the ideal of $p$-compact operators and contains the Banach ideal of all right $p$-nuclear holomorphic mappings. We also characterize holomorphic mappings with relatively $p$-compact ranges as those bounded holomorphic mappings which factorize through a quotient space of $\ell_{p^*}$ or as those whose transposes are quasi $p$-nuclear operators (respectively, factor through a closed subspace of $\ell_p$).