Essential Spectra, Matrix Operator and Applications

Abstract

This paper gives a survey on some recent results on the spectral theory of block operator matrices. We study the closeness in the product space and we give some conditions to characterize the essential spectra in the Browder resolvent set case. Furthermore, we apply the obtained results to several models such as delay equations, Sturm-Liouville problem and transport equations.

On the M-essential spectra of two-group transport equations

“…The purpose of this paper is to provide a new result, which allows us to investigate the M‐essential spectra of unbounded operator matrix under additive perturbations and to extend many known results in the literature (see for example ). Further, the obtained results are applied to discuss the M‐essential spectra of a one‐dimensional transport equation.…”

“…An essential role here is played by the M ‐spectra and the M ‐resolvent set, and all the results are new with respect to these spectral classes. The key tool is exploiting several results from perturbation theory and spectral theory, which allows us to formulate some supplements to many results presented in to obtain information about the M ‐essential spectra of the following two‐group transport equations, in the Banach space X : = L 1 ([ − a , a ] × [ − 1,1]; dxdv ), a > 0. where T , K , and M are defined by and with T 1 is the streaming operator with maximal domain, is the streaming operator with domain including the boundary condition related to the incoming and the outgoing flux by the boundary operator H , K 12 , K 21 , and K 22 are the collision operators, and M i ,1 ≤ i ≤ 4 are bounded operators (see Section 4 for more details).…”

Essential spectra of operator matrices and applications

“…Recently, some progress has been made by A. Jeribi et al in [9][10][11][12][13]. Furthermore, in [14][15][16], we find that an account research and a wide panorama of methods to investigate the spectral theory of block operator matrices are given.…”

On Relative Essential Spectra of Block Operator Matrices and Application

“…In this direction some issues may be found in the literature, we can quote for example [1,2,10,16,24]. Recently, an account research and a wide panorama of methods to investigate the spectral theory of block operator matrices is given in [3,4,5,11,14,25]. More precisely, the description of various essential spectra of a block operator matrix L appears in [3,5,11,14] to improve and generalize some results given by [1,2,24] for block operator matrices in Banach spaces under some compactness assumptions.…”

“…Recently, an account research and a wide panorama of methods to investigate the spectral theory of block operator matrices is given in [3,4,5,11,14,25]. More precisely, the description of various essential spectra of a block operator matrix L appears in [3,5,11,14] to improve and generalize some results given by [1,2,24] for block operator matrices in Banach spaces under some compactness assumptions. However, it should be noted that several results for the authors cited in the papers of [1,2,4,14,24] are aimed at providing methods for dealing with spectral theory for operator in the form L 0 − λM where M = I.…”