Gustafson, weidmann, kato, wolf, schechter and browder essential spectra of some matrix operator and application to two-group transport equation

Abstract

In this paper, we investigate the Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of a 2×2 block matrix operator defined on a Banach space where entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components under some properties. Furthermore, we give an application to two-group transport equation.

On Relative Essential Spectra of Block Operator Matrices and Application

“…More specific perturbations results are stated until the paper where they are used to describe the Fredholm, right and left Fredholm perturbations of the difference between the resolvents of two block operator matrices which ensure the stability on their M -essential spectra under weaker conditions than proved in the papers of [4,14,25]. All the results are new and…”

“…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.…”

“…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.…”

“…Moreover, the use of the M -resolvent, the Fredholm perturbations theory and the lower factorization allows us to formulate some supplements to many results presented in [4,14] and to ameliorate the description of the Messential spectra of two-group transport equations without knowing the totality of the relative essential spectra of the operator A, but only the relative essential spectra of its restriction which is more general than the one provided by [4,14]. The present paper consists of four sections: In Section 2, we present some basic notations and auxiliary lemmas connected to the main body of the paper.…”

Essential Spectra, Matrix Operator and Applications

“…In [14], A. Jeribi and I. Walha have studied the Gustafson, Weidmann, Kato, Wolf, Schechter, and Browder essential spectra of the matrix operator A (the closure of A 0 ) in the resolvent set of A 1 where A 1 := A | D(A)∩N (Γ X ) and N (Γ X ) is the kernel of Γ X .…”