Perturbations of operators similar to contractions and the commutator equation

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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⊕ V.