On similarity to contractions of class C· 0 with finite defects

A criterion on the similarity of a (bounded, linear) operator T on a (complex, separable) Hilbert space H in terms of shift-type invariant subspaces of T to a contraction of class C· 0 with finite unequal defects is given. Namely, T is similar to such a contraction if and only if the minimal quantity of (closed) invariant subspaces M of T such that the restriction T|M of T on M is similar to the simple unilateral shift, whose linear span is H, is finite. A sufficient condition for the similarity of an absolutely continuous polynomially bounded operator T to a contraction of class C· 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)= O, where B is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson--Newman product).