Spectral properties of non-selfadjoint difference operators

Let L denote the operator generated in l(2) (Z) by the difference expressio n (ly)(n) = a(n-1)y(n-1) + b(n)y(n) + a(n)y(n +1), n is an element of Z ={0 , +/- 1, +/- 2,...},where {a(n)}(n is an element of Z) and {b(n)}(n is an e lement of Z) are complex sequences. In this paper we investigated the spect rum, the spectral singularities, and the properties of the principal vector s corresponding to the spectral singularities of L. We also studied similar problems for the discrete Dirac operator M generated in l(2)(Z,C-2)by the system of difference expression (Lambday)(n) = (((Lambda 1y)n)((Lambda 2y)n)) = ((Delta yn(2) + pnyn(1))(-D elta yn-1(1) + qnyn(2))), n is an element of Z, where y = {((yn(1))(yn(2)))} (n is an element of Z) Delta is the forward differen ce operator, i.e., Deltay(n)((i)) = Delta ((i))(n+1) - y(n)((i)), = 1, 2, a nd {p(n)}(n is an element of Z), are complex sequences. (C) 2001 Academic P ress.