Cohen strongly p-summing holomorphic mappings on Banach spaces

Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\le p\le \mathrm{\infty }$. 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 ${\mathcal{G}}^{\mathrm{\infty }}(U)$ (the canonical predual of ${\mathcal{H}}^{\mathrm{\infty }}(U)$) and their transpositions on ${\mathcal{H}}^{\mathrm{\infty }}(U)$. Concerning the space ${\mathcal{D}}_p^{{\mathcal{H}}^{\mathrm{\infty }}}(U,F)$ formed by such mappings and endowed with a natural norm $d_p^{{\mathcal{H}}^{\mathrm{\infty }}}$, 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 $({\mathcal{D}}_p^{{\mathcal{H}}^{\mathrm{\infty }}}(U,F^{\ast }),d_p^{{\mathcal{H}}^{\mathrm{\infty }}})$ with the dual of the completion of tensor product space ${\mathcal{G}}^{\mathrm{\infty }}(U)\otimes F$ endowed with the Chevet–Saphar norm $g_p$.