Analytic equivalence and similarity of operators

The analytic equivalence of two operators is a generalization of similarity. We prove that under some conditions the analytic equivalence between two Hilbert space operators T and R implies the similarity of their restrictions on generalized ranges. We also prove that, in certain cases, the similarity of T to a contraction implies that of R. An improvement of a well-known criterion of similarity to an isometry due to Sz.-Nagy is given and an extension of a result of Apostol is obtained.