Cohen strongly p-summing holomorphic mappings on Banach spaces

TL;DR

This paper introduces and studies Cohen strongly p-summing holomorphic mappings between Banach spaces, establishing their properties, factorization theorems, and duality relations, thus extending the theory of summing operators to a holomorphic setting.

Contribution

It defines Cohen strongly p-summing holomorphic mappings, proves their key theorems, and characterizes their Banach ideal structure and duality properties.

Findings

Established Pietsch domination and factorization theorems for these mappings.

Identified the space as a regular Banach ideal generated by strongly p-summing linear operators.

Connected the dual space with tensor products endowed with Chevet--Saphar norm.

Abstract

Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\leq p\leq\infty$. We introduce and study the notion of a Cohen strongly $p$-summing holomorphic mapping from $U$ to $F$, a holomorphic version of a strongly $p$-summing linear operator. For such mappings, we establish both Pietsch domination/factorization theorems and analyse their linearizations from $\mathcal{G}^\infty(U)$ (the canonical predual of $\mathcal{H}^\infty(U)$) and their transpositions on $\mathcal{H}^\infty(U)$. Concerning the space $\mathcal{D}_p^{\mathcal{H}^\infty}(U,F)$ formed by such mappings and endowed with a natural norm $d_p^{\mathcal{H}^\infty}$, we show that it is a regular Banach ideal of bounded holomorphic mappings generated by composition with the ideal of strongly $p$-summing linear operators. Moreover, we identify the space $(\mathcal{D}_p^{\mathcal{H}^\infty}(U,F^*),d_p^{\mathcal{H}^\infty})$ with the dual of the completion of tensor product space $\mathcal{G}^\infty(U)\otimes F$ endowed with the Chevet--Saphar norm $g_p$.