Mean ergodicity of weighted composition operators on spaces of holomorphic functions

Abstract

[EN] Let phi be a self-map of the unit disc D of the complex plane C and let psi be a holomorphic function on D. We investigate the mean ergodicity and power boundedness of the weighted composition operator C-phi,C-psi(f) = psi(f o phi) with symbol phi and multiplier psi on the space H(D). We obtain necessary and sufficient conditions on the symbol phi and on the multiplier psi which characterize when the weighted composition operator is power bounded and (uniformly) mean ergodic. One necessary condition is that the symbol phi has a fixed point in D. If phi is not a rational rotation, the sufficient conditions are related to the modulus of the multiplier on the fixed point of phi. Some of our results are valid in an open connected set U of the complex plane.