Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations

Abstract

In the present paper, we extend in the first place the main results of Tretter in (Spectral theory block operator matrices and applications. Imperial College Press, London, 2008) to linear relations. Moreover, we define a matrix linear relation. We denote by $$\mathcal {L}$$ the block matrix linear relation, acting on the Banach space $$X\oplus Y$$ , of the form $$\begin{aligned} \mathcal {L}= \left( \begin{array}{ll} A &{} B\\ C &{} D \\ \end{array} \right) , \end{aligned}$$ where A, B, C and D are four closable linear relations with dense domains. For diagonally dominant and off-diagonally dominant of block matrix linear relation $$\mathcal {L}$$ , we give a necessary and sufficient condition for $$\mathcal {L}$$ to become closed and closable. In the second place, we study the stability of the essential spectrum of this block linear relation.