Spectral stability for compact perturbations of Toeplitz matrices

Let f   be a regular real-valued non-constant symbol defined on the one dimensional torus TT. Denote respectively by κ and T  , its set of critical points and the associated Toeplitz matrix on l2(N)l2(N). If V   is a suitable compact perturbation, we prove that the operator T+VT+V has no singular continuous spectrum and only finite point spectrum away from the set of thresholds f(κ)f(κ). We also obtain some propagation estimates and apply these results to concrete examples.