Weyl's theorem and tensor products: a counterexample
type: Item
instance of: scholarly article
title: Weyl's theorem and tensor products: a counterexample
zbMATH Open document ID: 1217.47008
author: Derek Kitson
author: Robin Harte
author: Carlos Hernández
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-03-14T00:00:00Z
review text: For a bounded operator \(T \in B(X)\) on a Banach space \(X\), denote by \(\omega_{\text{ess}}(T)\) and \(\beta_{\text{ess}}(T)\) its Weyl spectrum and its Browder spectrum, respectively. The operator is said to satisfy Weyl's theorem if the complement in the spectrum of the Weyl spectrum coincides with the isolated eigenvalues of finite multiplicity. In particular, such an operator also satisfies Browder's theorem, that is, \(\omega_{\text{ess}}(T)=\beta_{\text{ess}}(T)\). The authors construct an example of two operators \(A,B \in B(\ell_2)\) which satisfy Weyl's theorem each, and therefore also Browder's theorem, but whose tensor product \(A \otimes B \in B(\ell_2 \otimes \ell_2)\) does not satisfy Browder's theorem. The counterexample comprises the tensor product of two direct sums, each of which involves forward and backward shifts as simple building blocks. As a byproduct, the authors find that the product of circles in the complex plane is a limaçon.
reviewed by: Carsten Michels
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A80
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 5865642
zbMATH Keywords: Weyl's theorem
zbMATH Keywords: Browder's theorem
zbMATH Keywords: tensor product
zbMATH Keywords: shifts
zbMATH Keywords: spectral picture
zbMATH Keywords: limaçon
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.12.051
OpenAlex ID: W2088850594
cites work: WEYL'S THEOREM, TENSOR PRODUCTS AND MULTIPLICATION OPERATORS II
cites work: Q3961339
cites work: Another note on Weyl’s theorem
cites work: Weyl's theorem, tensor products and multiplication operators
Wikidata QID: Q124877131
DOI: 10.1016/J.JMAA.2010.12.051
description: scientific article; zbMATH DE number 5865642