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Bergman and Bloch spaces of vector-valued functions

✍ Scribed by José Luis Arregui; Oscar Blasco

Publisher
John Wiley and Sons
Year
2003
Tongue
English
Weight
265 KB
Volume
261-262
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

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~(𝔻, 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 ℬ︁(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 𝓁~p~(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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