On similarity to contraction semigroups in Hilbert space

It is proved that if T(t) is a bounded, uniformly continuous, quasi-compact C-0-semigroup in Hilbert space, then T(t) is similar to a contraction semigroup. Analogously, if T is a power-bounded quasi-compact operator, then T is similar to a contraction. This latter fact is a simultaneous generalization of well known results of Sz.-Nagy and Rota. Some characterizations of bounded quasi-compact semigroups and power-bounded quasi-compact operators are also given.