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

Abstract

In this paper, we deal with the M‐essential spectra of unbounded linear operators in Banach spaces where some generalizations of earlier work are given. Furthermore, we give an application from transport theory. Copyright © 2013 John Wiley & Sons, Ltd.

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…”

“…. , 6 (see Section 4 in [25] for more details) with Eq. (2), we obtain the following result for the M j -essential right and left spectra for j = 1, 4 of the operators T 1 and S λ as: which ends this proof.…”

“…Y ) into a Banach space Z. The closure of the block operator matrix L 0 , L := L 0 , is discussed in details in the paper of [25] under some assumptions on the entries components. The study of the block operator matrix is the subject of many authors under different assumptions.…”

“…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.…”

On the essential spectra of coupled operators matrix and application

“…Proof Let , then and Since is a relatively weak compact subset of L 1 ((−1,1), d ξ ), and K 12 defines a regular operator, then it follows from Lemma in Walha that K 21 ( λ − A ) −1 and ( λ − A ) −1 K 12 are 2 weak compact operators on X 1 . According to Lemma , one has , , and Then, we infer that and …”