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We prove that if alpha(t)=(1-t)(alpha) over tilde (t), where (alpha) over tilde has no zeros in [0,1], then T is similar to a contraction. Operators of this type have been investigated by Agler, Muller, Olofsson, Pott and others, however, we treat cases where their techniques do not apply. We write down an explicit Nagy-Foias type model of an operator in this class and discuss its usual consequences (completeness of eigenfunctions, similarity to a normal operator, etc.). We also show that the limits of parallel to T(n)h parallel to as n -> infinity, h is an element of H, do not exist in general, but do exist if an additional assumption on alpha is imposed. 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