Hyperinvariant subspaces and extended eigenvalues
type: Item
instance of: scholarly article
title: Hyperinvariant subspaces and extended eigenvalues
zbMATH Open document ID: 1090.47003
published in: The New York Journal of Mathematics
publication date: 2005-05-09T00:00:00Z
full work available at URL: https://eudml.org/doc/126140
full work available at URL: 10-5nf.htm
review text: For a bounded operator \(A\) on a complex Hilbert space \(H\), the set \(EE(A)\) of extended eigenvalues for \(A\) is defined to be the set of those complex numbers \(\lambda\) for which there is an operator \(T\neq 0\) satisfying \(AT=\lambda TA\). For a given \(\lambda\in EE(A)\), define \({\mathcal E}={\mathcal E}(A,\lambda)\) as the set of all \(\lambda\) eigen-operators for \(A\). In this paper, several scenarios are investigated for which the existence of non-unimodular extended eigenvalues leads to invariant or hyperinvariant subspaces. Some of the results are as follows: (1) Let \(\lambda\in EE(A)\) and let \({\mathcal E}={\mathcal E}(A,A)\). Define \(S= \{x:{\mathcal E}x= \{0\}\}\). Then \(S\) in hyperinvariant for \(A\). (2) Suppose that \(EE(A)\) is not closed under multiplication. Then \(A\) has a proper hyperinvariant subspace. (3) Suppose that \(A\) is invertible and \(T\in{\mathcal E}(A,\lambda)\), and \(\lambda\) is not a root of unity. Then the spectrum of \(T\) must be circularly symmetric. (4) Let \(\gamma\) be a nonzero scalar. Then \(EE(\gamma+ V)= \{1\}\), where \(V\) is the classical Volterra operator on \(L^2(0, 1)\).
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A62
zbMATH DE Number: 2166314
zbMATH Keywords: extended eigenvalues
zbMATH Keywords: hyperinvariant subspaces
author: Alan Lambert
reviewed by: Bahmann Yousefi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2166314