Power bounded composition operators on spaces of analytic functions

Abstract

We study the dynamical behaviour of composition operators defined on spaces of real analytic functions. We characterize when such operators are power bounded, i.e. when the orbits of all the elements are bounded. In this case this condition is equivalent to the composition operator being mean ergodic. In particular, we show that the composition operator is power bounded on the space of real analytic functions on if and only if there is a basis of complex neighbourhoods U of such that the operator is an endomorphism on the space of holomorphic functions on each U .

A note on mean ergodic composition operators on spaces of holomorphic functions

“…The evaluation of these equivalent conditions and the precise form of the projection which appears as limit of the Cesaro sums is determined in concrete cases in Corollaries 2 and 3. Our results here are utilized in [7] to investigate power bounded composition operators on spaces of real analytic functions.…”

A Note on Supercyclic Operators in Locally Convex Spaces

“…For instance, when ϕ is an automorphism. We can find similar examples in spaces of real analytic functions; see, e.g., [7,Corollary 2.5].…”

Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

“…The composition operator is definitely one of the most natural linear operators of analysis and there is an extensive literature on that subject: see the monographs in case of spaces of holomorphic functions [19,39] and the papers on a real analytic case [13,14,[20][21][22][23][24]. For a literature on the space of real analytic functions see a recent survey [20].…”