  Sz.-Nagy's famous theorem states that a bounded operator $T$ which acts on a
complex Hilbert space $\mathcal{H}$ 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
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.
