Perturbation of spectra of operator matrices and local spectral theory
type: Item
instance of: scholarly article
title: Perturbation of spectra of operator matrices and local spectral theory
zbMATH Open document ID: 1105.47006
author: E. H. Zerouali
author: H. Zguitti
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: Let \(X\) and \(Y\) be Banach spaces and let \(L(X,Y)\) denote the space of all bounded linear operators from \(X\) to \(Y\) (\(L(X):=L(X,X)\)). For \(A\in L(X)\), \(B\in L(Y)\) and \(C\in L(X,Y)\), denote by \(M_{C}\) the operator defined on \(X\oplus Y\) by \(\left[\begin{smallmatrix} A & C\\ 0 & B \end{smallmatrix} \right]\). In this article, the defect set \(D^{\Sigma }=(\Sigma (A)\cup \Sigma (B))\setminus \Sigma (M_{C})\) is studied for different spectra including the spectrum, the essential spectrum, the Weyl spectrum and the approximate point spectrum. The obtained results are applied to the stability of such spectra (\(D^{\Sigma }=\varnothing\)) and the classes of operators \(C\) for which stability holds of \(M_{C}\) using local spectral theory.
reviewed by: Bilender P. Allahverdiev
Mathematics Subject Classification ID: 47A11
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5073000
zbMATH Keywords: defect set
zbMATH Keywords: local spectral theory
zbMATH Keywords: operator matrices
zbMATH Keywords: spectra
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.065
OpenAlex ID: W2048805091
cites work: Operators which do not have the single valued extension property
cites work: Analytical Functional Models and Local Spectral Theory
cites work: A note on the spectrum of an upper triangular operator matrix
cites work: A duality theorem for an arbitrary operator
cites work: WEYL'S THEOREM, $a$-WEYL'S THEOREM, AND LOCAL SPECTRAL THEORY
cites work: Q4422033
cites work: A note on Weyl’s theorem for operator matrices
cites work: Perturbation of Spectrums of 2 × 2 Operator Matrices
cites work: Local spectral theory for \(2\times 2\) operator matrices
cites work: The single valued extension property on a Banach space
cites work: Invertible completions of $2\times 2$ upper triangular operator matrices
cites work: Local spectral properties of a square matrix of operators
cites work: The boundedness below of \(2\times 2\) upper triangular operator matrices
cites work: Asymptotic intertwining and spectral inclusions on Banach spaces
cites work: Weyl spectra of operator matrices
cites work: On the Weyl spectrum. II
cites work: Q3887989
cites work: The operator equation AX-XB=C with normal A and B
cites work: Similarity Orbits and the Range of the Generalized Derivation X →MX - XN
cites work: Hyponormal operators
cites work: On the weak decomposition property (δw)
DOI: 10.1016/J.JMAA.2005.12.065
description: scientific article; zbMATH DE number 5073000