Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions
type: Item
instance of: scholarly article
title: Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions
zbMATH Open document ID: 1108.47014
author: Nedra Moalla
author: Mondher Damak
author: Aref Jeribi
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: Let \(X\) be a Banach space, \({\mathcal L}(X)\) be the set of all bounded linear operators on \(X\), and \({\mathcal C}(X)\) be the set of all closed densely defined linear operators on \(X\). Let \(\Phi_-(X)\) and \(\Phi_+(X)\) be the sets of lower and upper semi-Fredholm operators in \({\mathcal C}(X)\) and \(\Phi(X)=\Phi_-(X)\cap\Phi_+(X)\) be the set of all Fredholm operators on \(X\). The authors study the following essential spectra of \(T\in{\mathcal C}(X)\): the Gustafson--Weidmann essential spectra \(\sigma_{e1}:= \{\lambda\in {\mathbb C}:\lambda-T\notin\Phi_+(X)\}\), \(\sigma_{e2}:=\{\lambda\in {\mathbb C}:\lambda-T\notin\Phi_-(X)\}\); the Kato essential spectrum \(\sigma_{e3}:=\{\lambda\in {\mathbb C}:\lambda-T\notin\Phi_-(X) \cup\Phi_+(X)\}\); the Wolf essential spectrum \(\sigma_{e4}:=\{\lambda\in {\mathbb C}:\lambda-T\notin\Phi(X)\}\); the Schechter essential spectrum \(\sigma_{e5}(T):= {\mathbb C}\setminus\rho_5(T)\), where \(\rho_5(T):=\{\lambda\in{\mathbb C}: \lambda-T\in\Phi(X),\;\text{ind}(\lambda-T)=0\}\); and the Browder essential spectrum \(\sigma_{e6}(T):= {\mathbb C}\setminus\rho_6(T)\), where \(\rho_6(T)\) denotes the set of those \(\lambda\in\rho_5(T)\) such that all scalars near \(\lambda\) belong to the resolvent \(\rho(T)\). An operator \(F\in{\mathcal L}(X)\) is called a Fredholm perturbation (resp., an upper semi-Fredholm perturbation, a lower semi-Fredholm perturbation) if \(U+F\in\Phi(X)\) (resp., \(U+F\in\Phi_+(X)\), \(U+F\in\Phi_-(X)\)) whenever \(U\in\Phi(X)\) (resp., \(U\in\Phi_+(X)\), \(U\in\Phi_-(X)\)). The set of all Fredholm (lower semi-Fredholm, upper semi-Fredholm) perturbations is denoted by \({\mathcal F}(X)\) (resp., \({\mathcal F}_-(X)\), \({\mathcal F}_+(X)\)). Suppose that \(L_0= \left(\begin{matrix} A & B \\ C & D \end{matrix}\right)\) is defined on \(X\times X\), where \(A\) is closed densely defined linear operator on \(X\) with nonempty resolvent set \(\rho(A)\); \(B\) is densely defined linear operator on \(X\) …
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5077890
zbMATH Keywords: closable operator
zbMATH Keywords: semi-Fredholm operator
zbMATH Keywords: strictly singular operator
zbMATH Keywords: strictly cosingular operator
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: Gustafson-Weidmann essential spectra
zbMATH Keywords: Kato essential spectrum
zbMATH Keywords: Wolf essential spectrum
zbMATH Keywords: Schechter essential spectrum
zbMATH Keywords: Browder essential spectrum
zbMATH Keywords: semi-Fredholm perturbation
zbMATH Keywords: two-group transport operator
reviewed by: Oleksiy Karlovych
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.10.080
OpenAlex ID: W2048413300
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DOI: 10.1016/J.JMAA.2005.10.080
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