Spectral properties of non-selfadjoint difference operators

Let L denote the operator generated in l(2) (Z) by the difference expression (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 element of Z) are complex sequences. In this paper we investigated the spectrum, the spectral singularities, and the properties of the principal vectors 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))(-Delta 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 difference operator, i.e., Deltay(n)((i)) = Delta ((i))(n+1) - y(n)((i)), = 1, 2, and {p(n)}(n is an element of Z), are complex sequences. (C) 2001 Academic Press.