Equivalence up to a rank one perturbation
type: Item
instance of: scholarly article
title: Equivalence up to a rank one perturbation
zbMATH Open document ID: 1032.47011
author: Alexei Poltoratski
published in: Pacific Journal of Mathematics
publication date: 2000-12-03T00:00:00Z
review text: The author studies the problem of unitary equivalence up to a rank one operator for unitary operators with simple spectra in Hilbert space. He gives a complete solution in the case of operators with pure singular spectrum which fills the whole unit circle. The operators \(U_1\) and \(U_2\) are said to be completely non-equivalent if there are no non-trivial invariant subspaces \(H_1\) and \(H_2\) for \(U_1\) and \(U_2\), respectively, such that the restriction of \(U_1\) on \(H_1\) is unitary equivalent to the restriction of \(U_2\) on \(H_2\). The authors shows that for completely non-equivalent singular cyclic unitary operators \(U\) and \(V\) such that \(\sigma(U)=\sigma(V)=T\) there exist a unitary operator \(W\) and a rank one operator \(K\) such that \(WUW^*=V+K\). Some function theoretic consequences of this result are presented. The argument is based on the Krein-Lifshits spectral shift for a unitary pair.
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 1537821
zbMATH Keywords: unitary operators
zbMATH Keywords: singular measures
zbMATH Keywords: spectral shift function
zbMATH Keywords: unitary equivalence
zbMATH Keywords: Cauchy and Poisson integrals
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: pjm.2000.194.175
OpenAlex ID: W1990435605
DOI: 10.2140/PJM.2000.194.175
description: scientific article; zbMATH DE number 1537821