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 Ω if and only if there is a basis of complex neighbourhoods U of Ω such that the operator is an endomorphism on the space of holomorphic functions on each U.
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Acknowledgments
The research of Bonet was partially supported by MEC and FEDER Project MTM2007-62643 and by GV Project Prometeo/2008/101. The research of Domański was supported in years 2007-2010 by Ministry of Science and Higher Education, Poland, Grant no. NN201 2740 33. The authors are very indebted to F. Bracci for providing some information concerning complex dynamics. The second named author is very grateful to colleagues from Valencia for warm hospitality during his stays there.
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Dedicated to the memory of our friend Klaus D. Bierstedt.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bonet, J., Domański, P. Power bounded composition operators on spaces of analytic functions. Collect. Math. 62, 69–83 (2011). https://doi.org/10.1007/s13348-010-0005-9
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DOI: https://doi.org/10.1007/s13348-010-0005-9
Keywords
- Spaces of real analytic functions
- Real analytic manifold
- Composition operator
- Mean ergodic operator
- Power bounded operator
- Orbit
- Hypercyclic operator
- Hyperbolic spaces