Fully summingmultilinear and holomorphicmappings into Hilbert spaces

It is known that whenever E 1 , … , E n are infinite dimensional L ∞ ‐spaces and F is any infinite dimensional Banach space, there exists a bounded n ‐linear mapping from E 1 × … × E n into F that fails to be absolutely (1; 2)‐summing. In this paper we generalize a theorem of S. Kwapién and obtain a sufficient condition in order to assure that a given n ‐linear mapping T from infinite dimensional L ∞ ‐spaces into an infinite dimensional Hilbert space is absolutely (1; 2)‐summing. Besides, we also give a sufficient condition in order to obtain a fully (1; 1)‐summing multilinear mapping from l 1 ×…× l 1 × l 2 into an infinite dimensional Hilbert space. In the last section we introduce the concept of fully summing holomorphic mappings and give the first examples of this kind of map. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)