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

TL;DR

This paper establishes a sufficient condition under which a polynomially bounded operator is similar to a contraction, involving invariant subspaces and Blaschke product conditions, with implications for operator theory.

Contribution

It provides a new criterion linking polynomial boundedness, invariant subspaces, and Blaschke products to operator similarity to contractions.

Findings

Operator T is similar to a contraction under specified conditions.

Polynomial boundedness cannot be replaced by power boundedness, as shown by Le Merdy's example.

The result connects invariant subspace properties with Blaschke product conditions.

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.