Extensions, dilations and functional models of dirac operators

Allahverdiev B.

INTEGRAL EQUATIONS AND OPERATOR THEORY, vol.51, no.4, pp.459-475, 2005 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 51 Issue: 4
  • Publication Date: 2005
  • Doi Number: 10.1007/s00020-003-1241-0
  • Title of Journal : INTEGRAL EQUATIONS AND OPERATOR THEORY
  • Page Numbers: pp.459-475

Abstract

A space of boundary values is constructed for minimal symmetric Dirac operator in the Hilbert space L-A(2) ((-infinity, infinity); C-2) with defect index (2,2) (in Weyl's limit-circle cases at +/-infinity). A description of all maximal dissipative (accretive), selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at +/-infinity. We investigate two classes of maximal dissipative operators with separated boundary conditions, called 'dissipative at -infinity' and 'dissipative at +infinity'. In each of these cases we construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix. We construct a functional model of the maximal dissipative operator and define its characteristic function. We prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.