  Let $T$ be a polynomially bounded operator, and let $\mathcal M$ be its
invariant subspace. Suppose that $P_{\mathcal M^\perp}T|_{\mathcal M^\perp}$ is
similar to a contraction, while $\theta(T|_{\mathcal M})=0$, where $\theta$ is
a finite product of Blaschke products with simple zeros satisfying the Carleson
interpolating condition (a Carleson--Newman Blaschke product). Then $T$ is
similar to a contraction. It is mentioned that Le Merdy's example shows that
the assumption of polynomially boundedness cannot be replaced by the assumption
of power boundedness.
