Compression of quasianalytic spectral sets of cyclic contractions
type: Item
instance of: scholarly article
title: Compression of quasianalytic spectral sets of cyclic contractions
zbMATH Open document ID: 1269.47009
author: László Kérchy
author: Vilmos Totik
published in: Journal of Functional Analysis
publication date: 2012-11-13T00:00:00Z
review text: Let \(\mathcal{H}\) be an infinite-dimensional separable Hilbert space. For \(T\) a bounded linear operator on \(\mathcal{H}\), recall that a closed subspace \(W\subset\mathcal{H}\) is a hyperinvariant subspace of \(T\) if \(W\) is invariant under any operator commuting with \(T\). We denote by Hlat\((T)\) the hyperinvariant subspace lattice of \(T\). In the paper under review, the authors consider the class \(\mathcal{L}_0(\mathcal{H})\) of cyclic quasianalytic contractions, and the subclass \(\mathcal{L}_{1}(\mathcal{H})\subset\mathcal{L}_0(\mathcal{H})\) containing operators whose quasianalytic spectral sets are the unit circle. It is known by the work of the first author [J. Funct. Anal. 246, No. 2, 281--301 (2007; Zbl 1123.47008)] that every operator in \(\mathcal{L}_{1}(\mathcal{H})\) has a rich invariant subspace lattice. The main result of the present paper asserts that for every operator \(T\in\mathcal{L}_0(\mathcal{H})\), there exists an operator \(T_1\in\mathcal{L}_1(\mathcal{H})\) commuting with \(T\). It then follows that the identity Hlat\((T) =\) Hlat\((T_1)\) holds. As a consequence, the Hyperinvariant Subspace Problem (HSP) in the class \(\mathcal{L}_0(\mathcal{H})\) is equivalent to the HSP in the class \(\mathcal{L}_1(\mathcal{H})\). The operator \(T_1\) in the main theorem is given by \(T_1=f(T)\), where \(f\) is an appropriate \(H^{\infty}\)-function on the unit disk. The existence of such an \(f\) is proved by using tools from potential theory.
reviewed by: Trieu Le
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A45
zbMATH DE Number: 6104942
zbMATH Keywords: quasianalytic spectral set
zbMATH Keywords: hyperinvariant subspace
zbMATH Keywords: equilibrium measure
zbMATH Keywords: absolute continuity
MaRDI profile type: Publication
full work available at URL: j.jfa.2012.08.016
OpenAlex ID: W2089988052
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DOI: 10.1016/J.JFA.2012.08.016
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description: scientific article; zbMATH DE number 6104942