Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces
type: Item
instance of: scholarly article
title: Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces
zbMATH Open document ID: 0531.47029
DOI: 10.1007/BF01691914
author: Domingo A. Herrero
published in: Integral Equations and Operator Theory
publication date: 1983-01-01T00:00:00Z
review text: An injective bilateral weighted shift T on \(\ell^ 2\) is unitarily equivalent to multiplication by the variable on a space \(L^ 2(\beta)\) of formal Laurent series. By means of this aspect the following question raised by B. Chevreau is investigated: If \(\sigma_ p(T^*)\) is a circle, can one conclude that f is not in the closed linear span of \(\{T^ nf:n=1,2,...\}\) for any \(f\neq 0?\) One result is, that in case \(L^ 2(\beta)\) consists only of quasi- analytic functions on the boundary \(\partial D\) of the unit disc, then \(\partial D\subset \sigma_ p(T^*)\), and all cyclic invariant subspaces \({\mathcal M}\) have the property \(\cap \{T^ n{\mathcal M}:n=0,1,2,...\}=\{0\}\), and hence Chevreau's conclusion holds. Some corollaries and examples are given and it is demonstrated, that the condition of quasi-analyticity is essential. [For part I see Revista Un. Math. Argentina 26, 24-41 (1972; Zbl 0249.47024)].
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 3842550
zbMATH Keywords: space of formal Laurent series
zbMATH Keywords: quasi-analytic functions on the boundary of the unit disc
zbMATH Keywords: injective bilateral weighted shift
MaRDI profile type: Publication
cites work: Q3921721
cites work: Harmonic analysis based on certain commutative Banach algebras
cites work: Q5615632
cites work: Q5730397
cites work: Q5662952
cites work: Q5644622
cites work: On multicyclic operators
cites work: Q5556050
cites work: Q4059478
description: scientific article; zbMATH DE number 3842550