SIMILARITY TO A CONTRACTION, FOR POWER-BOUNDED OPERATORS WITH FINITE PERIPHERAL SPECTRUM

Suppose T is a power-bounded linear opertor on a Hilbert space with fi nite peripheral spectrum (spectrum on the unit circle). Several suffic ient conditions are given for T to be similar to a contraction. A natu ral growth condition on the resolvent in half-planes tangent to the un it circle at the peripheral spectrum is shown to be equivalent to T ha ving an H-infinity(P)boolean AND C((P) over bar) functional calculus, for some open polygon 'P contained in the unit disc, which, in turn, i s equivalent to T being similar to a contraction with numerical range contained in a closed polygon in the closed unit disc. Having certain orbits of T be square summable also implies that T is similar to a con traction.