Generalized Toeplitz operators and cyclic vectors

We give in this paper some asymptotic Von Neumann inequalities for power bounded operators in the class C subrho intersection C sub 1. and some spacial von Neumann inequalities associated with non zero elements of the point spectrum, when it is non void, of generalized Toeplitz operators. Introducing perturbed kernel, we consider classes C sub R which extend the classical classes C subrho. We give results about absolute continuity with respect to the Haar measure for operators in class C sub R intersection C sub 1. This allows us to give new results on cyclic vectors for such operators and provides invariant subspaces for their powers. Relationships between cyclic vectors for T and T* involving generalized Toeplitz operators are given and the commutativity of left brace T right brace', the commutant of T is discussed.