ASYMPTOTIC LIMITS OF OPERATORS SIMILAR TO NORMAL OPERATORS.

Sz.-Nagy's famous theorem states that a bounded operator T which acts on a complex Hilbert space H is similar to a unitary operator if and only if T is invertible and both T and T-1 are power bounded. There is an equivalent reformulation of that result which considers the self-adjoint iterates of T and uses a Banach limit L. In this paper first we present a generalization of the necessity part in Sz.-Nagy's result concerning operators that are similar to normal operators. In the second part we provide a characterization of all possible strong operator topology limits of the self-adjoint iterates of those contractions which are similar to unitary operators and act on a separable infinite-dimensional Hilbert space. This strengthens Sz.-Nagy's theorem for contractions. [ABSTRACT FROM AUTHOR]