Mathematische Zeitschrift

, Volume 261, Issue 3, pp 649–657

Dynamics of the differentiation operator on weighted spaces of entire functions

Article

Abstract

The continuity of the differentiation operator on weighted Banach spaces of entire functions with sup-norm has been characterized recently by Harutyunyan and Lusky. We give necessary and sufficient conditions to ensure that the differentiation operator on these weighted Banach spaces of entire functions is hypercyclic or chaotic, when it is continuous.

Mathematics Subject Classification (2000)

Primary 47A16 Secondary 46E15 47B38 

Keywords

Weighted spaces of entire functions Differentiation operator Hypercyclic operator Chaotic operator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bayart, F., Grivaux, S.: Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94, 181–210 (2007)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bayart, F., Matheron, É.: Hypercyclic operators which do not satisfy the hypercyclitity criterion. J. Funct. Anal. 250, 426–441 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bermúdez, T., Bonilla, A., Peris, A.: On hypercyclicity and supercyclicity criteria. Bull. Aust. Math. Soc. 70, 45–54 (2004)MATHGoogle Scholar
  5. 5.
    Bernal-González, L., Bonilla, A.: Exponential type of hypercyclic entire functions. Arch. Math. 78, 283–290 (2002)MATHCrossRefGoogle Scholar
  6. 6.
    Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Mich. Math. J. 40, 271–297 (1993)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Studia Math. 127, 137–168 (1998)MATHMathSciNetGoogle Scholar
  9. 9.
    Blasco, O., Bonilla, A., Grosse-Erdmann, K.G.: Rate of growth of frequently hypercyclic functions. Preprint (2007)Google Scholar
  10. 10.
    Bonet, J.: Hypercyclic and chaotic convolution operators. J. Lond. Math. Soc. 62, 253–262 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bonilla, A., Grosse-Erdmann, K.G.: Frequently hypercyclic operators and vectors. Ergod. Theory Dyn. Syst. 27, 383–404 (2007)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 64, 101–118 (1998)MATHGoogle Scholar
  13. 13.
    Costakis, G., Sambarino, M.: Topologically mixing hypercyclic operators. Proc. Am. Math. Soc. 132, 385–389 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison Wesley, Reading (1989)MATHGoogle Scholar
  15. 15.
    De La Rosa, M., Read, Ch.: A hypercyclic operator whose direct sum TT is not hypercyclic. J. Oper. Theory (to appear)Google Scholar
  16. 16.
    Godefroy, G., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Grivaux, S.: Hypercyclic operators, mixing operators and the bounded steps problem. J. Oper. Theory 54, 147–168 (2005)MATHMathSciNetGoogle Scholar
  18. 18.
    Grosse-Erdmann, K.G.: On the universal functions of G.R. MacLane. Complex Var. Theory Appl. 15, 193–196 (1990)MATHMathSciNetGoogle Scholar
  19. 19.
    Grosse-Erdmann, K.G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. 36, 345–381 (1999)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Grosse-Erdmann, K.G.: Rate of growth of hypercyclic entire functions. Indag. Math. (N.S.) 11, 561–571 (2000)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Grosse-Erdmann, K.G.: Hypercyclic and chaotic weighted backward shifts. Studia Math. 139, 47–68 (2000)MATHMathSciNetGoogle Scholar
  22. 22.
    Grosse-Erdmann, K.G.: Recent developments in hypercyclicity. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97, 273–286 (2003)MATHMathSciNetGoogle Scholar
  23. 23.
    Grosse-Erdmann, K.G.: A weak criterion for vector-valued holomorphy. Math. Proc. Cambridge Philos. Soc. 136, 399–411 (2004)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Harutyunyan, A, Lusky, W: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)MATHMathSciNetGoogle Scholar
  26. 26.
    Lusky, W.: On the Fourier series of unbounded harmonic functions. J. Lond. Math. Soc. 61, 568–580 (2000)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    MacLane, G.R.: Sequences of derivatives and normal families. J. Anal. Math. 2, 72–87 (1952/1953)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Martínez-Giménez, F., Peris, A.: Chaos for backward shift operators. Int. J. Bifur. Chaos Appl. Sci. Eng. 12, 1703–1715 (2002)MATHCrossRefGoogle Scholar
  30. 30.
    Salas, H.: Supercyclicity and weighted shifts. Studia Math. 135, 55–74 (1999)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Pura y Aplicada IUMPA, Edificio IDI5 (8E), Cubo F, Cuarta PlantaUniversidad Politécnica de ValenciaValenciaSpain

Personalised recommendations