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Journal of Operator Theory

Volume 49, Issue 1, Winter 2003  pp. 3-23.

Operators near completely polynomially dominated ones and similarity problems

Summary:  Let $T$ and $C$ be two Hilbert space operators. We prove that if $T$ is near, in a certain sense, to an operator completely polynomially dominated with a finite bound by $C$, then $T$ is similar to an operator which is completely polynomially dominated by the direct sum of $C$ and a suitable weighted unilateral shift. Among the applications, a refined Banach space version of Rota similarity theorem is given and partial answers to a problem of K. Davidson and V. Paulsen are obtained. The latter problem concerns CAR-valued Foguel-Hankel operators which are generalizations of the operator considered by G. Pisier in his example of a polynomially bounded operator not similar to a contraction.