The paper under review deals with spectral properties of block operator matrices on a direct sum of Banach spaces. The investigated problems are important because many complicated boundary value problems and systems of differential equations can be written in this form. The author investigates the essential spectrum of such operators using the so called Feshbach maps (also called Schur complements in the literature), which are obtained after a diagonalization procedure. Making suitable assumptions, the Fredholm property of these maps is investigated and used to locate the essential spectrum of the original matrix operator. The results are then applied to a matrix composed of a \(2n\)-th order differential operator and multiplication operators.