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Dilation and functional model of dissipative operator generated by an infinite Jacobi matrix. (English) Zbl 1047.47023
Summary: We consider the maximal dissipative operators acting in the Hilbert space ${\ell }_{ℂ}^{2}\left(ℕ;E\right)$ $\left(ℕ=\left\{0,1,2,\cdots \right\}$, $dimE=n<\infty$) that are extensions of a minimal symmetric operator with maximal deficiency indices $\left(n,n\right)$ (in the completely indeterminate case or the 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, making 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 completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.
##### MSC:
 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations 47A20 Dilations, extensions and compressions of linear operators 47A40 Scattering theory of linear operators 47A45 Canonical models for contractions and nonselfadjoint operators