Cyclicity of bicyclic operators
type: Item
instance of: scholarly article
title: Cyclicity of bicyclic operators
zbMATH Open document ID: 1116.47007
author: Aharon Atzmon
author: Sophie Grivaux
author: Evgeny Abakumov
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2007-05-04T00:00:00Z
review text: Given an injective operator on a separable Banach space \(X\), let \(R^\infty(T)\) be the intersection of the ranges of all operators \(T^n\) (\(n\geq 1\)). The operator \(T\) is called \textit{bicyclic} if there is \(x_0\in R^\infty(T)\) such that the linear span of \(\{T^nx_0:\) \(n\in{\mathbb Z}\}\) is dense in \(X\). If there is a vector \(x_0\in X\) such that the linear span of the orbit \(\{T^nx_0:\) \(n\geq0\}\) is dense in \(X\), the operator \(T\) is said to be \textit{cyclic}. In the paper under review, the authors prove, under suitable conditions on the norms \(\| T^nx_0\| \) (\(n\in{\mathbb Z}\)), that a bicyclic operator is, in fact, cyclic. As an application, the authors characterize a special class of weights \(\omega\) such that the backward shift operator is cyclic on \(\ell^2_\omega(\mathbb Z)\). In addition, they also establish that if \(S\) is a bounded invertible shift on \(\ell^2_\omega(\mathbb Z)\), under some condition on the asymptotic behavior of \(\omega\), either \(S\) or \(S^*\) is cyclic, thus answering, in the affirmative, a question proposed by \textit{D.\,A.\thinspace Herrero} [Rev.\ Unión Mat.\ Argent.\ 26, 24--41 (1972; Zbl 0249.47024)] in the non-quasianalytic case. The paper concludes with some results on completeness of translates in certain Banach spaces of real functions.
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 5149407
zbMATH Keywords: bicyclic operator
zbMATH Keywords: cyclic operator
zbMATH Keywords: shift operator
reviewed by: José A. Prado-Bassas
MaRDI profile type: Publication
full work available at URL: j.crma.2007.02.008
OpenAlex ID: W2048722371
cites work: Operators which are annihilated by analytic functions and invariant subspaces
cites work: Generating sets for Beurling algebras
cites work: Subnormal operators
cites work: Completeness in \(L^1(\mathbb R)\) of discrete translates
cites work: Universal families and hypercyclic operators
cites work: Q5667131
cites work: Q5662952
cites work: Q4110074
cites work: Q4059478
cites work: Q3667152
DOI: 10.1016/J.CRMA.2007.02.008
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description: scientific article; zbMATH DE number 5149407