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Bergman and Bloch spaces of vector‐valued functions
Author(s) -
Arregui José Luis,
Blasco Oscar
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310109
Subject(s) - mathematics , bergman space , bergman kernel , bloch space , banach space , projection (relational algebra) , space (punctuation) , lp space , dual space , bounded function , pure mathematics , normed vector space , unit sphere , mathematical analysis , linguistics , philosophy , algorithm
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)