Dilation, Model, Scattering and Spectral Problems of Second-Order Matrix Difference Operator

In the Hilbert space l(Omega)(2) (Z; E) (Z := {0,+/- 1,+/- 2,...}, dim E = N < infinity), the maximal dissipative singular second-order matrix difference operators that the extensions of a minimal symmetric operator with maximal deficiency indices (2N, 2N) (in limit-circle cases at +/- infinity) are considered. The maximal dissipative operators with general boundary conditions are investigated. For the dissipative operator, a self-adjoint dilation and is its incoming and outgoing spectral representations are constructed. These constructions make it possible to determine the scattering matrix of the dilation. Also a functionalmodel of the dissipative operator is constructed. Then its characteristic function in terms of the scattering matrix of the dilation is set. Finally, a theorem on the completeness of the system of root vectors of the dissipative operator is proved.