Geodesic
A sufficient condition for the similarity of a polynomially bounded operator to a contraction
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 77-95 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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M. F. Gamal'. A sufficient condition for the similarity of a polynomially bounded operator to a contraction. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 77-95. http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a5/

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