  Let $T$ and $V$ be two Hilbert space contractions and let $X$ be a linear
bounded operator. It was proved by C. Foias and J.P. Williams that in certain
cases the operator block matrix $R(X;T,V)$ (defined in the text) is similar to
a contraction if and only if the commutator equation $X = TZ-ZV$ has a bounded
solution $Z$. We characterize here the similarity to contractions of some
operator matrices $R(X;T,V)$ in terms of growth conditions or of perturbations
of $R(0;T,V)=T\oplus V$.
