Abstract: In this study, we examine some spectral properties
of non-selfadjoint matrix-valued difference equations consisting of
a polynomial-type Jost solution. The aim of this study is to
investigate the eigenvalues and spectral singularities of the difference
operator L which is expressed by the above-mentioned difference
equation. Firstly, thanks to the representation of polynomial type Jost
solution of this equation, we obtain asymptotics and some analytical
properties. Then, using the uniqueness theorems of analytic functions,
we guarantee that the operator L has a finite number of eigenvalues
and spectral singularities.