Bounded analytic functions in the Dirichlet space
type: Item
instance of: scholarly article
title: Bounded analytic functions in the Dirichlet space
zbMATH Open document ID: 0628.32005
DOI: 10.1007/BF01163287
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173917
review text: The Dirichlet space D(G) of a connected open set G in the complex plane is the set of all those analytic functions in G whose first derivative is square integrable with respect to area measure on G. The authors show that each such function can be represented as the quotient of two bounded functions in D(G). This result is used to study the structure of the (closed) subspaces of D(G) that are invariant under the algebra of multiplication operators on the Dirichlet space, in case G is simply connected. It is shown that every nontrivial invariant subspace contains a nontrivial bounded function, and that each two nontrivial invariant subspaces have a nontrivial intersection. When G is the unit disk it is shown that if M is an invariant subspace then zM has codimension one in M. This is in contrast to the situation for the Bergman space (the area square-integrable analytic functions in G) where all these results are known to be false, when G is the unit disk.
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 4021568
zbMATH Keywords: Dirichlet space
zbMATH Keywords: unit disk
zbMATH Keywords: bounded analytic functions
author: Stefan Richter
author: Allen L. Shields
MaRDI profile type: Publication
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OpenAlex ID: W2008354797
description: scientific article; zbMATH DE number 4021568