Abstract. Let $ R$ be a power bounded operator, and let $\cal M$ be its invariant subspace such that $ R|_\cal M$ is similar to a unilateral shift of finite multiplicity and $P_\cal M^\perp R|_\cal M^\perp $ is similar to a $C_0$-contraction. Then $R$ is similar to a contraction.
AMS Subject Classification
(1991): 47A45, 47A65, 47B99
power bounded operator,
Received February 21, 2014, and in final form September 1, 2014. (Registered under 17/2014.)