On similarity to contractions of class $C_{\cdot 0}$ with finite defects

Abstract

A criterion on the similarity of a (bounded, linear) operator $T$ on a (complex, separable) Hilbert space $\mathcal H$ in terms of shift-type invariant subspaces of $T$ to a contraction of class $C_{\cdot 0}$ 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 $\mathcal M$ of $T$ such that the restriction $T|_{\mathcal M}$ of $T$ on $\mathcal M$ is similar to the simple unilateral shift, whose linear span is $\mathcal H$, is finite. 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).