Spectral Properties Of Anti-Triangular Operator Matrices

The anti-triangular operator matrix is naturally arises from some special class of second-order linear differential equations,and plays an important role in real applica-tions.At present,the authors mainly focus on the characterization of spectra and some related properties for anti-triangular operator matrices.This paper studies the spectral properties of anti-triangular operator matrices.Firstly,the relations between the closed range spectrum,left spectrum,left(right)essential spectrum,essential spectrum,left(right)Weyl spectrum of the operator matrix and the corresponding properties of its op-erator entries are given by means of space decomposition.Then,the essential spectrum and Weyl spectrum are studied by Schur decomposition,and the relations between the properties of the operator matrix and those of its Schur complement are obtained.