On holomorphic mappings with relatively $p$-compact range

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 $\ell_p$).