Extensions, dilations and functional models of infinite Jacobi matrix

Allahverdiev B.

CZECHOSLOVAK MATHEMATICAL JOURNAL, vol.55, no.3, pp.593-609, 2005 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 55 Issue: 3
  • Publication Date: 2005
  • Doi Number: 10.1007/s10587-005-0048-3
  • Page Numbers: pp.593-609


A space of boundary values is constructed for the minimal symmetric operator generated by an infinite Jacobi matrix in the limit-circle case. A description of all maximal dissipative, accretive and selfadjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We construct a functional model of the dissipative operator and define its characteristic function. We prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative operators.