Essential Spectra of 2 × 2 Block Operator Matrices

Abstract

Let X and Y be Banach spaces. In the product space X Y , we consider an operator formally defined by a matrix and acts from Y into X (resp. from X into Y ). One of the problems in the study of such operators is that in general L 0 is not closed or even closable, even if its entries are closed. The aim of this chapter is to present some hypotheses which should allow the block operator matrix L 0 to be closable. For this purpose, it is interesting to present the suitable conditions for the entries A, B, C , and D. These conditions are mainly based on the FrobeniusSchur decomposition of L 0 . Then, our reasoning will be mainly based on the specific properties of the first entry A, hence allowing us to get a closable block matrix operator L 0 . It is interesting to notice that this reasoning based on A could also be applied for the other entries, namely B, C , or D. In fact our approach is based on three situations. In the first one, the resolvent of the operator A is a Fredholm perturbation. In the second situation, the operator A is closed, whereas this same operator is closable in the third situation. After that, we will give the essential spectra of the closure of the operator L 0 .