Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions

Abstract

In this article we investigate the essential spectra of a 2 × 2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to determine the essential spectra of two-group transport operators with general boundary conditions in the Banach space L p ([−a, a] × [−1, 1]) × L p ([−a, a] × [−1, 1]), a > 0.

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

“…2). Thus, Moalla et al [19] extend the obtained results into a large class of operators and describe many essential spectra of A and they apply their results to describe the essential spectra of two-group transport operators with general boundary conditions in L p -spaces. But, to determine the essential spectra of A, they must absolutely know the one of the entry A of the matrix (1.1).…”

“…Indeed, the use of the Browder resolvent and the lower-upper factorization given by [18] allow us to formulate and give some supplements to many results presented in [2]. By comparison with the papers of [4,19], we mention that we can determine the essential spectra of matrix A without having the essential spectra of the operator A, but we know only the one of its restriction A 1 and we will give in our work an application in transport theory which is more general than the one provided in [19].…”

Essential Spectra of a 3 × 3 Operator Matrix and an Application to Three-Group Transport Equations

“…Since C\σ ew (A 1 ), C\σ ew (S 1 (μ)), C\σ ew (S 2 (μ)) and C\σ ew (L) are connected and ρ(S 1 (μ)), ρ(S 2 (μ)) and ρ(L) are not empty, the result follows from [5, Proposition 2.3] together with [20,Theorem 3.2].…”

“…Moreover, the compactness of the operators (λ−A) −1 (see [2]) or C(λ−A) −1 and ((λ − A) −1 B) * (see [29]) for some (and hence for all) λ ∈ ρ(A) was assumed, whereas in [6], it was only assumed that (λ − A) −1 for λ ∈ ρ(A) belongs to a non-zero two-sided closed ideal I(X) ⊂ F(X) of L(X). In [20], Moalla, Damak and Jeribi extended these results to a large class of operators, described their essential spectra, and applied these results to describe the essential spectra of two-group transport operators with general boundary conditions in L p -spaces. In [16], Jeribi, Moalla and Walha treated a 3 × 3 block operator matrix (1.1) on a Banach space.…”

“…We focus on the investigation of the closability and the description of the essential spectra. Compared with the papers [6,16,20], we can determine the essential spectra of the closure of (1.1) without knowing the essential spectra of the operator A but only that of one of its restrictions A 1 , and we give an application to transport theory which is more general than the one considered in [16]. In fact, in the Banach space 1]; dxdξ), a > 0, we consider an operator that describes the neutron transport in a plane-parallel domain with width 2a, or the transfer of unpolarized light in a plane-parallel atmosphere of optical thickness 2a (see Sect.…”