Abstract
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.
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Kumar, M., Srivastava, S. Convergence of powers of composition operators on certain spaces of holomorphic functions defined on the right half plane. Arch. Math. 110, 487–500 (2018). https://doi.org/10.1007/s00013-018-1159-z
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DOI: https://doi.org/10.1007/s00013-018-1159-z