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.

In this article, we give some results on the S‐essential spectra of a linear operator defined on a Banach space. Furthermore, we apply the obtained results to determine the S‐essential spectra of an integro‐differential operator with abstract boundary conditions in the Banach space Lp([−a,a] × [−1,1]),p ≥ 1 and a > 0. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper a new concept for a 3 × 3 block operator matrix is studied on a Banach space.It is shown that, under certain conditions, it defines a closable operator and its essential spectra are determined. Application to transport operators in L 1 -space is given.

This paper is devoted to the investigation of the spectral stability of unbounded operator matrices with non diagonal domain in product of Banach spaces. Our results are aimed to characterize some essential spectra of this kind of operators in terms of the union of the essential spectra of the restriction of its diagonal operators entries. The abstract results are illustrated by an example of two-group transport equations with perfect periodic boundary conditions.