On composition ideals and dual ideals of bounded holomorphic mappings

Applying a linearization theorem due to J. Mujica, we study the ideals of bounded holomorphic mappings ${\mathcal{H}}^{\mathrm{\infty }}\circ \mathcal{I}$ 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 ${\mathcal{I}}^{\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}}$. 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 $1\le p<\mathrm{\infty }$. We prove that every $p$-integral ($p$-nuclear) holomorphic mapping from $U$ to $F$ has relatively weakly compact (compact) range.