A Sufficient Condition for the Similarity of a Polynomially Bounded Operator to a Contraction

Abstract

Let T be a polynomially bounded operator and let ℳ be its invariant subspace. Assume that PM⊥T |M⊥\( {\left.{P}_{{\mathrm{\mathcal{M}}}^{\perp }}T\right|}_{{\mathrm{\mathcal{M}}}^{\perp }} \) is similar to a contraction, while θ(T|_ℳ) = 0, where θ is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson–Newman product). Then T is similar to a contraction. It is mentioned that Le Merdy’s example shows that the assumption of polynomial boundedness cannot be replaced by the assumption of power boundedness.