Abstract
We characterize those composition operators defined on spaces of holomorphic functions of several variables which are power bounded, i.e. the orbits of all the elements are bounded. This condition is equivalent to the composition operator being mean ergodic. We also describe the form of the symbol when the composition operator is mean ergodic.
Article PDF
Similar content being viewed by others
References
Abate M.: Iteration Theory of Holomorphic Maps on Taut Manifolds. Mediterranean Press, Commenda di Rende (1989)
Albanese, A.A., Bonet, J., Ricker, W.J.: On mean ergodic operators. In: Curbera, G.P. et al. (eds.) Vector Measures, Integration and Related Topics, vol. 201, pp. 1–20. Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (2009)
Albanese A.A., Bonet J., Ricker W.J.: Mean ergodic operators in Fréchet spaces. Anal. Acad. Sci. Fenn. Math. 34, 401–436 (2009)
Bayart, F., Matheron, E.: Dynamics of linear operators. In: Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)
Bernal-González L., Montes-Rodríguez A.: Universal functions for composition operators. Complex Var. 27, 47–56 (1995)
Bernal-González L.: Universal entire functions for affine endomorphisms of \({\mathbb{C}^{\mathbb{N}}}\). J. Math. Anal. Appl. 305, 690–697 (2005)
Bonet, J., Domański, P.: Power bounded composition operators on spaces of analytic functions. Collect. Math. doi:10.1007/s13348-010-0005-9 (2010)
Cowen, C.C., MacCluer, B.D.: Composition operators on spaces of analytic functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)
Fonf V.P., Lin M., Wojtaszczyk P.: Ergodic characterizations of reflexivity in Banach spaces. J. Funct. Anal. 187, 146–162 (2001)
Grosse-Erdmann K.G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. 36, 345–381 (2001)
Grosse-Erdmann K.G.: Recent developments in hypercyclicity. Rev. R. Acad. Cien. Ser. A Math. 97, 273–286 (2003)
Grosse-Erdmann K.G., Mortini R.: Universal functions for composition operators with non-automorphic symbol. J. Anal. Math. 107, 355–376 (2009)
Grosse-Erdmann K.G, Peris A.: Linear Chaos. Springer, Berlin (2011)
Gunning R.C.: Introduction to Holomorphic Functions of Several Variables. Wadsworth & Brooks, Belmont (1990)
Hörmander L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland, Amsterdam (1990)
Kobayashi S.: Hyperbolic Complex Spaces. Springer, Berlin (1998)
Krengel U.: Ergodic Theorems. de Gruyter, Berlin (1985)
Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)
Milnor J.: Dynamics in One Complex Variable. Vieweg Verlag, Wiesbaden (2000)
Shapiro, J.H.: Composition operators and classical function theory. In: Universitext: Tracts in Mathematics. Springer, New York (1993)
Shapiro, J.H.: Notes on the Dynamics of Linear Operators. Lecture notes. http://www.math.msu.edu/~shapiro/Pubvit/Downloads/LinDynamics/lindynamics.pdf (1993)
Yosida K.: Functional Analysis. Springer, Berlin (2001)
Acknowledgments
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.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of Bonet was partially supported by MEC and FEDER Projects MTM2007-62643 and MTM2010-15200 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.
Rights and permissions
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.
About this article
Cite this article
Bonet, J., Domański, P. A note on mean ergodic composition operators on spaces of holomorphic functions. RACSAM 105, 389–396 (2011). https://doi.org/10.1007/s13398-011-0009-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-011-0009-7