On a characterization of the essential spectra of some matrix operators and application to two-group transport operators

Abstract

In this paper, we investigate the essential approximate point spectrum and the essential defect spectrum of a 2×2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to two-group transport operators in the Banach space

Essential Spectra, Matrix Operator and Applications

“…Theorem 3.1 [5] Assume that (H1)-(H8) hold. Then the block operator matrix A 0 is closable in X × Y if and only if the operator M λ := D + CK λ Γ Y − C λ B is closable in Y and its closure A is given by the relation…”

“…(iii) It is proven in [5] that for λ ∈ ρ b (A 1 ) the restriction of the operator Γ λ := Γ X | N (A λ ) is injective and its range, denoted by Z 1 , does not depend on λ, where A λ is the operator defined on D(A) by…”

“…Our approach to this problem was introduced by S. Charfi and A. Jeribi in [5] to study the essential defect spectrum and the essential approximate point spectrum. Among the works in this direction we quote, for example, [4,14,21].…”

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.…”