Dynamics of weighted composition operators on weighted Banach spaces of entire functions

Abstract

We study the dynamics of the weighted composition operator Cw,ϕ on the weighted Banach spaces of entire functions Hv(C) and H0 v (C). We characterize the continuity and compactness of the operator and, in the case of affine symbols ϕ(z) = az+b, a, b ∈ C, and exponential weights, we analyze when the operator is power bounded, (uniformly) mean ergodic and hypercyclic. Continuous weighted composition operators when |a| = 1 are just multiples of composition operators λCϕ λ ∈ C. When |a| < 1, we consider as a multiplier w the product of a polynomial by an exponential function. For multiples of composition operators, we get a complete characterization of power boundedness and mean ergodicity and we study the hypercyclicity in terms of λ. An example of a power bounded but not mean ergodic operator on H0 v (C) is provided. For the case of composition operators, we obtain the spectrum and a complete characterization of the dynamics.