A criterion on the similarity of a (bounded, linear) operator T on a (complex, separable) Hilbert space \({\mathcal {H}}\) to a contraction of class \(C_{\cdot 0}\) with finite unequal defects is given in terms of shift-type invariant subspaces of T. Namely, T is similar to such a contraction if and only if there exists a finite collection of (closed) invariant subspaces \({\mathcal {M}}\) of T such that the restriction \(T|_{{\mathcal {M}}}\) of T on \(\mathcal M\) is similar to the simple unilateral shift and the linear span of these subspaces \({\mathcal {M}}\) is \({\mathcal {H}}\) . A sufficient condition for the similarity of an absolutely continuous polynomially bounded operator T to a contraction of class \(C_{\cdot 0}\) with finite equal defects is given. Namely, T is similar to such a contraction if the (spectral) multiplicity of T is finite and \(B(T)=\mathbb O\) , where B is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson–Newman product).
