A sufficient condition for the similarity of a polynomially bounded operator to a contraction

Abstract

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.

Translated from Zapiski nauchn. semin. POMI, v. 456, 2017