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

M. F. Gamal'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 77-95

Résumé

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. 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.