Bergman and Bloch spaces of vector‐valued functions

We investigate Bergman and Bloch spaces of analytic vector‐valued functions in the unit disc. We show how the Bergman projection from the Bochner‐Lebesgue space L p (, X ) onto the Bergman space B p ( X ) extends boundedly to the space of vector‐valued measures of bounded p ‐variation V p ( X ), using this fact to prove that the dual of B p ( X ) is B p ( X *) for any complex Banach space X and 1 < p < ∞. As for p = 1 the dual is the Bloch space ℬ( X *). Furthermore we relate these spaces (via the Bergman kernel) with the classes of p ‐summing and positive p ‐summing operators, and we show in the same framework that B p ( X ) is always complemented in p ( X ). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)