On holomorphic mappings with relatively p-compact range

Abstract

Related to the concept of p-compact operators with p ? [1,?] introduced by Sinha and Karn [20], this paper deals with the space H? Kp (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 H?(U) are p-compact operators. This fact allows us to make a complete study of them. We show that H? Kp 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 ?p* or as those whose transposes are quasi p-nuclear operators (respectively, factor through a closed subspace of ?p).