  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 polynomial bounded operator not similar to a contraction.
