Dilation and functional model of dissipative operator generated by an infinite jacobi matrix

We consider the maximal dissipative operators icting in the Hilbert space lc2(N;E) (N = {0,1, 2, … ∼, dim E = n < ∞) that the extensions of a minimal symmetric operator with maximal deficiency indices (n, n) (in completely indeterminate case or limit-circle case) generated by an infinite Jacobi matrix with matrix entries. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the scattering matrix of the dilation. We prove the theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators. 2003 Elsevier Ltd. All rights reserved.