Linear resolvent growth test for similarity of a weak contraction to a normal operator
type: Item
instance of: scholarly article
title: Linear resolvent growth test for similarity of a weak contraction to a normal operator
zbMATH Open document ID: 1031.47003
DOI: 10.1007/BF02388793
author: Stanislav Kupin
published in: Arkiv för Matematik
publication date: 2003-09-30T00:00:00Z
review text: In order to describe the main result of the paper under review, consider a complex separable Hilbert space \(H\) and a bounded linear operator \(T\) on \(H\) with \(\|T\|\leq 1\). Denote by \({\mathbf D}\) the open unit disc in the complex plane and note that for any \(\mu\in{\mathbf D}\) one can define the bounded linear operator \(b_\mu(T)=(1-\bar\mu T)^{-1}(T-\mu)\). Then the main theorem of the present paper says that \(T\) is similar to a normal operator provided that the spectrum \(\sigma(T)\) is different from the closure of \({\mathbf D}\) and the following conditions are satisfied: \[ \sup_{\lambda\in{\mathbf C}\setminus\sigma(T)}\|(T-\lambda)^{-1}\|\text{dist} (\lambda,\sigma(T))<\infty \] and \[ \sup_{\mu\in{\mathbf D}}\text{tr} (1-b_\mu(T)^*b_\mu(T))<\infty, \] where \(\text{tr}(\cdot)\) stands for the usual trace supported by the ideal of trace-class operators on \(H\). The proof is accomplished firstly under the additional assumption that \(T\) has a complete set of eigenvectors. (A basic tool used at this stage is the Sz-Nagy-Foiaş model for completely nonunitary contractions.) The author then derives the result in the general case described above. We conclude by mentioning that the paper is clearly written and to a large extent self-contained.
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A45
zbMATH DE Number: 1987644
zbMATH Keywords: complex separable Hilbert space
zbMATH Keywords: spectrum
zbMATH Keywords: Sz-Nagy-Foiaş model
zbMATH Keywords: similarity to a normal operator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1987644