Spectral Theory of Second-Order Vector Difference Equations*

This paper is concerned with spectral problems of second-order vector difference equations with two-point boundary value conditions, where the matrix-valued coefficient of the leading term may be singular. A concept of self-adjointness of the boundary value conditions is introduced. The self-adjointness of the corresponding difference operator is discussed on a suitable admissible function space, and fundamental spectral results are obtained. The dual orthogonality of eigenfunctions is shown in a special case. Rayleigh’s principles and the minimax theorems in two linear spaces are given. As an application, a comparison theorem for eigenvalues of two Sturm]Liouville problems is presented