Our purpose is to investigate the asymptotic properties of an operator T on an invariant subspace E 2 Lat(T) and on E? with the generalized Toeplitz operators associated with T. We show how the relative properties may be used in order to give a general result linking the behaviour of T on E and on E? with the possibility for T to be similar to a scalar multiple of a contraction. Some applications are indicated. In particular, one of our results implies that there is no hope to construct a power bounded operator of Foguel type that is not similar to a contraction and such that for everyx in H-{0} the sequence (Tn) does not converge to 0. We also study the asymptotic and spectral properties of these operators of Foguel type.