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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access November 20, 2014

Dynamics of differentiation operators on generalized weighted Bergman spaces

  • Liang Zhang and Ze-Hua Zhou EMAIL logo
From the journal Open Mathematics


The chaos of the differentiation operator on generalized weighted Bergman spaces of entire functions has been characterized recently by Bonet and Bonilla in [CAOT 2013], when the differentiation operator is continuous. Motivated by those, we investigate conditions to ensure that finite many powers of differentiation operators are disjoint hypercyclic on generalized weighted Bergman spaces of entire functions.


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[1] Bernal-González L., Disjoint hypercyclic operators, Studia Math., 2007, 182(2), 113-131. 10.4064/sm182-2-2Search in Google Scholar

[2] Bonet J., Dynamics of differentiation operator on weighted spaces of entire functions, Math. Z., 2009, 261, 649-657. 10.1007/s00209-008-0347-0Search in Google Scholar

[3] Bonet J., Bonilla A., Chaos of the differentiation operator on weighted Banach spaces of entire functions, Complex Anal. Oper. Theory, 2013, 7, 33-42. 10.1007/s11785-011-0134-5Search in Google Scholar

[4] Bermúdez T., Bonilla A., Conejero J. A., Peris A., Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math., 2005, 170, 57-75. 10.4064/sm170-1-3Search in Google Scholar

[5] Bonilla A., Grosse-Erdmann K. G., Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems, 2007, 27, 383-404. Erratum: Ergodic Theory Dynam. Systems, 2009, 29, 1993-1994. 10.1017/S0143385709000959Search in Google Scholar

[6] Bayart F., Matheron É., Dynamics of linear operators, Cambridge Tracts in Mathematics, 179, Camberidge University Press, Cambridge, 2009. 10.1017/CBO9780511581113Search in Google Scholar

[7] Bès J., Martin Ö., Peris A., Disjoint hypercyclic linear fractional composition operators, J. Math. Appl., 2011, 381, 843-856. 10.1016/j.jmaa.2011.03.072Search in Google Scholar

[8] Bès J., Martin Ö., Peris A., Shkarin S., Disjoint mixing operators, J. Funct. Anal., 2012, 263, 1283-1322. 10.1016/j.jfa.2012.05.018Search in Google Scholar

[9] Bès J., Martin Ö., Sanders R., Weighted shifts and disjoint hypercyclicity, 2012, manuscript. Search in Google Scholar

[10] Bès J., Peris A., Disjointness in hypercyclicity, J. Math. Anal. Appl., 2007, 336, 297-315. 10.1016/j.jmaa.2007.02.043Search in Google Scholar

[11] Costakis G., Sambarino M., Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc., 2004, 132(2), 385-389. 10.1090/S0002-9939-03-07016-3Search in Google Scholar

[12] Chen R. Y., Zhou Z. H., Hypercyclicity of weighted composition operators on the unit ball of CN, J. Korean Math. Soc., 2011, 48(5), 969-984. 10.4134/JKMS.2011.48.5.969Search in Google Scholar

[13] Grosse-Erdmann K. G., Peris Manguillot A., Linear Chaos, Springer, New York, 2011. 10.1007/978-1-4471-2170-1Search in Google Scholar

[14] Harutyunyan A., Lusky W., On the boundedness of the differentiation operator between weighted spaces of holomorphic functions, Studia Math., 2008, 184, 233-247. 10.4064/sm184-3-3Search in Google Scholar

[15] Lusky W., On generalized Bergman space, Studia Math., 1996, 119, 77-95. 10.4064/sm-119-1-77-95Search in Google Scholar

[16] Lusky W., On the Fourier series of unbounded harmonic functions, J. London. Math. Soc., 2000, 61, 568-580. 10.1112/S0024610799008443Search in Google Scholar

[17] Salas H. N., Dual disjoint hypercyclic operators, J. Math. Anal. Appl., 2011, 374, 106-117. 10.1016/j.jmaa.2010.09.003Search in Google Scholar

[18] Shkarin S., A short proof of existence of disjoint hypercyclic operators, J. Math. Anal. Appl., 2010, 367, 713-715.10.1016/j.jmaa.2010.01.005Search in Google Scholar

Received: 2013-6-15
Accepted: 2014-6-30
Published Online: 2014-11-20

© 2015 Liang Zhang and Ze-Hua Zhou,

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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