Spectral properties of singular discrete linear Hamiltonian systems

Abstract

This paper is concerned with spectral properties of singular discrete linear Hamiltonian systems. It is shown that properties of the essential spectrum of each self-adjoint subspace extension (SSE) of the corresponding minimal subspace are independent of the values of the coefficients of the system on any finite subinterval. The analyticity of the Weyl function is studied by employing the Schwarz reflection principle for the system in the limit point case. Based on the above work, several sufficient conditions are obtained for each SSE to have no essential spectrum points in an interval of the real line in the strong limit point case, and then a sufficient condition for the essential spectrum to be bounded from below (above) and a sufficient condition for the pure discrete spectrum are presented, respectively. As a direct consequence, the related spectral properties of singular higher order symmetric vector difference expressions are given.