Mean ergodicity of weighted composition operators on spaces of holomorphic functions

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