Extensions, dilations and functional models of discrete Dirac operators in limit point–circle cases

A space of boundary values is constructed for symmetric discrete Dirac operators in ℓ²A(Z; C²) (Z ≔ {0, ±1, ±2, …}) with defect index (1, 1) (in Weyl's limit‐circle case at ±∞ and limit‐point case at ∓∞). A description of all maximal dissipative (accretive), self‐adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with boundary conditions, called ‘dissipative at −∞’ and ‘dissipative at ∞’. In each of these cases we construct a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of the self‐adjoint operator. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative operators.