Approximate spectral synthesis in the Bergman space
type: Item
instance of: scholarly article
title: Approximate spectral synthesis in the Bergman space
zbMATH Open document ID: 0982.46021
DOI: 10.1215/S0012-7094-00-10111-1
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: Recall that a space of analytic functions on the unit disc is called \(z\)-invariant if it is invariant under the operator \(M\) of multiplication by the independent variable. For the Bergman space \(L^2_a\) it is proved that any \(z\)-invariant subspace of index \(1\) admits strong approximate spectral cosynthesis, i.e.\ it may be written as a lower limit of a sequence of finite-codimensional \(z\)-invariant subspaces such that also the annihilator is the lower limit of the corresponding annihilators. Moreover, any \(z^*\)-invariant subspace of the dual (i.e.\ invariant under \(M^*\)) admits weak approximate spectral synthesis, i.e.\ it may be written as a lower limit of a sequence of finite-dimensional \(z^*\)-invariant subspaces. This proves a conjecture of \textit{N. K. Nikol'skij} [J. Sov. Math. 26, 2185-2186 (1984; Zbl)]. The result has some applications to approximation by rational functions.
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 43A45
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 1425271
zbMATH Keywords: Bergman space
zbMATH Keywords: \(z\)-invariant subspace
zbMATH Keywords: approximate spectral synthesis
zbMATH Keywords: lower limit
zbMATH Keywords: rational approximation
zbMATH Keywords: spectral cosynthesis
zbMATH Keywords: annihilators
author: S. M. Shimorin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1425271