Completely indecomposable operators and a uniqueness theorem of Cartwright-Levinson type
type: Item
instance of: scholarly article
title: Completely indecomposable operators and a uniqueness theorem of Cartwright-Levinson type
zbMATH Open document ID: 0945.47012
author: Aharon Atzmon
author: M. I. Sodin
published in: Journal of Functional Analysis
publication date: 2000-10-04T00:00:00Z
review text: A completely indecomposable operator is a bounded linear operator \(T\) on some complex Hilbert space, such that the spectrum of \(T\) is not a singleton, is contained in the spectrum of the restriction of \(T\) to any nonzero invariant subspace and the analogous property holds for \(T^*\). For example, neither the unilateral shift, nor the bilateral one is completely indecomposable. The aim of the paper under review is to provide two classes of completely indecomposable operators. The first class consists of self-adjoint operators with spectrum \([-2,2]\) perturbed by certain compact quasinilpotent operators. At present it is not known whether or not these operators actually possess nontrivial invariant subspaces. The operators from the second class are perturbations of the bilateral shift by certain compact quasinilpotent operators and their spectrum is the unit circle. Concerning these operators, the question arises whether or not they possess nontrivial hyperinvariant subspaces. The proofs of complete indecomposability for the above mentioned operators involve local spectral theory, hence analytic extensions of the resolvent. The main technical ingredient here is a new uniqueness theorem of Cartwright-Levinson type, concerning analytic extension of holomorphic functions in the complement of the unit circle. We should finally note that, besides of its extremely important theorems, the paper also contains several open problems and additional discussions.
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 30D60
zbMATH DE Number: 1398377
zbMATH Keywords: weighted shift
zbMATH Keywords: weighted bi-shift
zbMATH Keywords: hyperdistribution
zbMATH Keywords: quasianalytic
zbMATH Keywords: local spectrum
zbMATH Keywords: invariant subspace
zbMATH Keywords: completely indecomposable operator
zbMATH Keywords: self-adjoint operators
reviewed by: Daniel Beltiţă
MaRDI profile type: Publication
full work available at URL: jfan.1999.3454
OpenAlex ID: W2076933832
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DOI: 10.1006/JFAN.1999.3454
description: scientific article; zbMATH DE number 1398377