Complex Analysis and Operator Theory

, Volume 7, Issue 1, pp 33–42 | Cite as

Chaos of the Differentiation Operator on Weighted Banach Spaces of Entire Functions



Motivated by recent work on the rate of growth of frequently hypercyclic entire functions due to Blasco, Grosse-Erdmann and Bonilla, we investigate conditions to ensure that the differentiation operator is chaotic or frequently hypercyclic on generalized weighted Bergman spaces of entire functions studied by Lusky, whenever the differentiation operator is continuous. As a consequence we partially complete the knowledge of possible rates of growth of frequently hypercyclic entire functions for the differentiation operator.


Weighted spaces of entire functions Differentiation operator Hypercyclic operator Chaotic operator Frequently hypercyclic operator 

Mathematics Subject Classification (2000)

Primary 47A16 Secondary 46E15 47B38 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bayart F., Grivaux S.: Hypercyclicité: le rôle du spectre ponctuel unimodulaire. C. R. Math. Acad. Sci. Paris 338, 703–708 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bayart F., Grivaux S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bayart F., Matheron E.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, Vol. 179. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  4. 4.
    Bernal-Gonz alez L., Bonilla A.: Exponential type of hypercyclic entire functions. Arch. Math. (Basel) 78, 283–290 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bierstedt K.D., Bonet J., Galbis A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40, 271–297 (1993)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Blasco O., Bonilla A., Grosse-Erdmann K.-G.: Rate of growth of frequently hypercyclic functions. Proc. Edinburgh Math. Soc. 53, 39–59 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bonilla A., Grosse-Erdmann K.-G.: On a theorem of Godefroy and Shapiro. Integr. Equa. Oper. Theory 56, 151–162 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007), 383–404. Erratum: Ergodic Theory Dynam. Systems 29 (6)(2009), 1993–1994Google Scholar
  9. 9.
    Bonet J.: Dynamics of the differentiation operator on weighted spaces of entire functions. Math. Z. 261, 649–657 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Grivaux, S.: A new class of frequently hypercyclic operators, with applications. to appear in Indiana Univ. Math. J.Google Scholar
  11. 11.
    Grosse-Erdmann K.-G.: On the universal functions of G. R. MacLane. Complex Variables Theory Appl. 15, 193–196 (1990)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Grosse-Erdmann K.-G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. (N.S.) 36, 345–381 (1999)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Grosse-Erdmann K.-G.: Rate of growth of hypercyclic entire functions. Indag. Math. (N.S.) 11, 561–571 (2000)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Grosse-Erdmann K.G.: Recent developments in hypercyclicity. Rev. R. Acad. Cien. Serie A Mat. 97, 273–286 (2003)MathSciNetMATHGoogle Scholar
  15. 15.
    Grosse-Erdmann K.G.: A weak criterion for vector-valued holomorphy. Math. Proc. Cambridge Philos. Soc. 136, 399–411 (2004)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Grosse-Erdmann, K.G., Peris, A.: Linear Chaos. Springer, Berlin (to appear)Google Scholar
  17. 17.
    Harutyunyan A., Lusky W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Stud. Math. 184, 233–247 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lusky W.: On generalized Bergman spaces. Stud. Math. 119, 77–95 (1996)MathSciNetMATHGoogle Scholar
  19. 19.
    Lusky W.: On the Fourier series of unbounded harmonic functions. J. Lond. Math. 61, 568–580 (2000)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lusky W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175, 19–45 (2006)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    MacLane G.R.: Sequences of derivatives and normal families. J. Anal. Math. 2, 72–87 (1952/53)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Murray J.D.: Asymptotic Analysis. Springer, New York (1984)MATHCrossRefGoogle Scholar
  23. 23.
    Shkarin S.A.: On the growth of D-universal functions. Univ. Math. Bull. 48(6), 49–51 (1993)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática, Pura y Aplicada IUMPAUniversidad Politécnica de ValenciaValenciaSpain
  2. 2.Departamento de Análisis MatemáticoUniversidad de La LagunaLa Laguna (Tenerife)Spain

Personalised recommendations