A note on the spectral singularities of non-selfadjoint matrix-valued difference operators (Q984911)

The author develops the spectral theory of block Jacobi matrices \(J=J\{A_n,B_n\}\), where \(A_n\), \(B_n\) are non-selfadjoint matrices of order \(m\), \(\det A_n\not=0\). The main results concern the class of \(J\) such that \[ \sum_{n=1}^\infty n(\|I-A_n\|+\|B_n\|)<\infty. \] The author proves that the continuous spectrum \(\sigma_c(J)=[-2,2]\), the set of eigenvalues is countable and its limit points lie on \([-2,2]\), the set of spectral singularities \(\sigma_{ss}(J)\subset [-2,2]\) and has Lebesgue measure zero. An example of a non-selfadjoint matrix difference operator with a spectral singularity is given.