Asymptotic Behaviour of the Powers of Composition Operators on Banach Spaces of Holomorphic Functions

We study the asymptotic behaviour of the powers T-n of a composition operator T on an arbitrary Banach space X of holomorphic functions on the open unit disc D of C. We show that for composition operators, one has the following dichotomy: either the powers converge uniformly or they do not converge even strongly. We also show that uniform convergence of the powers of an operator T is an element of L(X) is very much related to the behaviour of the poles of the resolvent of T on the unit circle of C, and that all poles of the resolvent of the composition operator T on X are algebraically simple. Our results are applied to study the asymptotic behaviour of semigroups of composition operators associated with holomorphic semiflows.