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 ≤ p ≤∞. 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 G∞ (U) (the canonical predual of H∞ (U)) and their transpositions on H∞ (U). Concerning the space DH∞ p formed by such mappings and endowed with a natural norm dH∞p, 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 (DH∞ p (U, F∗), dH∞p) with the dual of the completion of tensor product space G∞(U) ⊗ F endowed with the Chevet–Saphar norm gp.