Difference operators as semigroup generators

In this paper we examine difference operators with constant coefficients. We show that the type of the generated semigroup is determined by a matrix B, originating from the domain of the operator. Moreover, we provide necessary and sufficient conditions for exponential and polynomial stability of the semigroup in terms of the matrix B, using results using results of A. Borichev and Y. Tomilov. We close the paper with an application of our results to flows in networks.