Convergence of powers of composition operators on certain spaces of holomorphic functions defined on the right half plane

This paper studies the asymptotic behaviour of the powers (C_varphi ^n) of a composition operator (C_varphi ) on certain spaces of holomorphic functions defined on the right half plane (mathbb {C}_+). It is shown that for composition operators on the Hardy spaces and the standard weighted Bergman spaces, if the inducing map (varphi ) is not of parabolic type, then either the powers (C_varphi ^n) converge uniformly only to 0 or they do not converge even strongly.