Shift invariant subspaces with arbitrary indices in \(\ell^p\) spaces
type: Item
instance of: scholarly article
title: Shift invariant subspaces with arbitrary indices in \(\ell^p\) spaces
zbMATH Open document ID: 1016.47008
author: Alexander Borichev
author: Evgeny Abakumov
published in: Journal of Functional Analysis
publication date: 2002-07-04T00:00:00Z
review text: Let \(S(a_0,a_1,\ldots)=(0,a_0,a_1,\ldots)\) be the shift operator on \(\ell_2\). It follows from Beurling's theorem that \(\dim (M/SM)=1\) for every non-zero shift invariant subspace \(M\) of \(\ell_2\). In this paper, the authors consider \(S\) as an operator on some weighted \(\ell_p\) spaces, \(0<p<\infty\). The index of the invariant subspace \(M\) is \(\dim (M/\overline{SM})\). It is shown that, for \(2<p<\infty\) and for every \(n=2,3,\ldots,\infty\), there exist shift invariant subspaces of \(\ell_p\), or more generally of some weighted \(\ell_p\) spaces, with index \(n\). The proof is rather long and technical, and the authors begin with the case \(n=2\), without weight, which is simpler. They point out that the holomorphic functions naturally associated to the elements of the shift invariant subspaces they construct have no common zeros in the unit disk \({\mathcal D}\). The paper ends with several open questions.
reviewed by: Daniel Li
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 1763765
zbMATH Keywords: Banach spaces of weighted sequences
zbMATH Keywords: cyclic multiplicity
zbMATH Keywords: index of an invariant subspace
MaRDI profile type: Publication
full work available at URL: jfan.2001.3850
OpenAlex ID: W2033541608
cites work: Cyclicity and approximation by lacunary power series
cites work: Pseudocontinuations and the backward shift
cites work: Beurling's theorem for the Bergman space
cites work: Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I
cites work: Invariant subspaces of arbitrary multiplicity for the shift on \(\ell^1\)
cites work: On two problems concerning linear transformations in Hilbert space
cites work: Invariant subspaces of given index in Banach spaces of analytic functions
cites work: Completeness of translates in weighted spaces on the half-line
cites work: Sous-espaces invariants par translations bilatérales de certains espaces de Hilbert de suites quasi-analytiquement pondérées
cites work: An invariant subspace of the Bergman space having the codimension two property.
cites work: Linear and complex analysis problem book 3. Part 2
cites work: Invariant Subspaces in Banach Spaces of Analytic Functions
DOI: 10.1006/JFAN.2001.3850
description: scientific article; zbMATH DE number 1763765