Cohen strongly p-summing holomorphic mappings on Banach spaces

Abstract

Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\le p\le \infty$. 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 $({\mathcal{D}}_p^{{\mathcal{H}}^{\infty }}(U,F^{\ast }),d_p^{{\mathcal{H}}^{\infty }})$ with the dual of the completion of tensor product space ${\mathcal{G}}^{\infty }(U)\otimes F$ endowed with the Chevet--Saphar norm $g_p$.