Spectral Properties of Non-selfadjoint Difference Operators

AbstractLet L denote the operator generated in ℓ2(Z) by the difference expression (ℓy)n=an−1yn−1+bnyn+anyn+1,n∈Z={0,±1,±2,…}, where {an}n∈Z and {bn}n∈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 ℓ2(Z,C2) by the system of difference expressionΛyn=(Λ1y)n(Λ2y)n=Δy(2)n+pny(1)n−Δy(1)n−1+qny(2)n,n∈Z,wherey=y(1)ny(2)nn∈Z,Δ is the forward difference operator, i.e., Δy(i)n=y(i)n+1−y(i)n, i=1,2, and {pn}n∈Z,{qn}n∈Z are complex sequences