Invariant subspaces in Bergman spaces and the biharmonic equation
type: Item
instance of: scholarly article
title: Invariant subspaces in Bergman spaces and the biharmonic equation
zbMATH Open document ID: 0833.46044
author: Peter Duren
author: Dmitry Khavinson
author: H. S. Shapiro
author: Carl Sundberg
published in: Michigan Mathematical Journal
publication date: 1996-03-20T00:00:00Z
review text: The authors continue their research on contractive divisors in the Bergman space \(A^p (\Omega)\) of all \(p\)-integrable analytic functions on a bounded planar domain \(\Omega\). Results of \textit{H. Hedenmalm} [J. Reine Angew. Math. 422, 45-68 (1991; Zbl 0734.30040)]\ for the case \(p=2\) and the authors [Pac. J. Math. 157, 37-56 (1993; Zbl 0782.30027)]for \(1\leq p\leq \infty\) on \(z\)-invariant subspaces of \(A^p\) defined by \(A^p\) zero sets are generalized to \(A^p\)-spaces with \(0< p< 1\) as well as to arbitrary \(z\)-invariant subspaces of \(A^p (\mathbb{D})\). It is also shown that parts of the theory can be extended to Bergman spaces on simply connected Jordan domains with analytic boundary. The key tool of this theory is an integral formula involving the biharmonic Green function. The paper concludes with several interesting questions.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 682539
zbMATH Keywords: contractive divisors
zbMATH Keywords: \(z\)-invariant subspaces
zbMATH Keywords: Bergman spaces
zbMATH Keywords: Jordan domains with analytic boundary
zbMATH Keywords: biharmonic Green function
Wikidata QID: Q56672817
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004992
OpenAlex ID: W2091855704
DOI: 10.1307/MMJ/1029004992
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description: scientific article; zbMATH DE number 682539