Invariant subspaces for Bishop-type operators
type: Item
instance of: scholarly article
title: Invariant subspaces for Bishop-type operators
zbMATH Open document ID: 0727.47002
DOI: 10.1016/0022-1236(90)90146-C
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: The main focus of this paper is hyperinvariant subspaces for weighted translation operators on \(L^ 2[0,1]\). The author exhibits a rather large class including many operators of the form \(M_{\phi}U_{\alpha}\) on \(L_ 2[0,1]\), where \(\phi \in L^{\infty}[0,1]\) and \(U_{\alpha}\) is translation modulo one, by an irrational \(\alpha\). If for example \(\phi\) is the restriction to [0,1] of a function analytic in some open neighborhood of [0,1] in the complex plane, then \(M_{\phi}U_{\alpha}\) has a nontrivial hyperinvariant subspace for almost all \(\alpha\). This improves a result of \textit{A. M. Davie} [Bull. London Math. Soc. 6, 343- 348 (1974; Zbl 0287.47003)]. Originally operators of type \(M_ xU_{\alpha}\) have been suggested by Bishop as candidates for operators with no nontrivial subspace. The question however of whether \(M_ xU_{\alpha}\) has a nontrivial invariant subspace for all \(\alpha\) remains open.
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 4200967
zbMATH Keywords: Bishop-type operators
zbMATH Keywords: hyperinvariant subspaces for weighted translation operators on \(L^ 2[0,1]\)
author: Gordon Macdonald
reviewed by: Volker Wrobel
MaRDI profile type: Publication
cites work: Q3708592
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cites work: The existence of invariant subspaces
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full work available at URL: 0022-1236(90)90146-c
OpenAlex ID: W2032199755
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description: scientific article; zbMATH DE number 4200967