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.

We define an abstract setting to treat essential spectra of unbounded coupled operator matrix. We prove a well-posedness result and develop a spectral theory which also allows us to prove an amelioration to many earlier works. We point out that a concrete example from integro-differential equation fit into this abstract framework involving a general class of regular operator in L 1 spaces. KEYWORDScoupled operator matrix, essential spectra, Fredholm perturbations theory, perfect periodic boundary conditions, transport operator l, w, ss, b}, 7218

This paper deals with a new description of the one sided operator matrix form, as a generalization of the case of the unbounded operator matrix with the non diagonal domain, to investigate some advances in the analysis of some essential spectra under weaker hypotheses then the one provided in the works of [17, 33]. An example of differential equations is tested to ensure the validity of the abstract results.