  Let $f$ be a regular real-valued non-constant symbol defined on the one
dimensional torus ${\mathbb T}$. Denote respectively by $\kappa$ and $T$, its
set of critical points and the associated Toeplitz matrix on $l^2({\mathbb
N})$. If $V$ is a suitable compact perturbation, we prove that the operator
$T+V$ has no singular continuous spectrum and only finite point spectrum away
from the set of thresholds $f(\kappa)$. We also obtain some propagation
estimates and apply these results to concrete examples.
