Advertisement

Chaotic differential operators

  • J. Alberto Conejero
  • Félix Martínez-GiménezEmail author
Original Paper

Abstract

We give sufficient conditions for chaos of (differential) operators on Hilbert spaces of entire functions. To this aim we establish conditions on the coefficients of a polynomial P(z) such that P(B) is chaotic on the space \({\ell^p}\) , where B is the backward shift operator.

Keywords

Chaotic operators Hypercyclic operators Differential operators Backward shifts 

Mathematics Subject Classification (2000)

47A16 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Google Scholar
  2. 2.
    Bermúdez T., Miller V.G.: On operators T such that f(T) is hypercyclic. Integr. Equ. Oper. Theory 37(3), 332–340 (2000)CrossRefGoogle Scholar
  3. 3.
    Bonet J., Martínez-Giménez F., Peris A.: Linear chaos on Fréchet spaces. Int. J. Bifur. Chaos Appl. Sci. Eng. 13(7), 1649–1655 (2003)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chan K.C., Shapiro J.H.: The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40(4), 1421–1449 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Conejero J.A., Müller V.: On the universality of multipliers on \({\mathcal{H}({\mathbb {C}})}\). J. Approx. Theory. 162(5), 1025–1032 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    deLaubenfels R., Emamirad H.: Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory Dyn. Syst. 21(5), 1411–1427 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. In: Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989)Google Scholar
  8. 8.
    Godefroy G., Shapiro J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229–269 (1991)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Grosse-Erdmann K.-G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139(1), 47–68 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Grosse-Erdmann, K.-G., Peris, A.,: Linear chaos. Universitext, Springer, New York (to appear, 2011)Google Scholar
  11. 11.
    Herzog G., Schmoeger C.: On operators T such that f(T) is hypercyclic. Stud. Math. 108(3), 209–216 (1994)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kahane, J.-P.: Some random series of functions, 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge University Press, Cambridge (1985)Google Scholar
  13. 13.
    Martínez-Giménez F., Peris A.: Chaos for backward shift operators. Int. J. Bifur. Chaos Appl. Sci. Eng. 12(8), 1703–1715 (2002)zbMATHCrossRefGoogle Scholar
  14. 14.
    Martínez-Giménez F.: Chaos for power series of backward shift operators. Proc. Am. Math. Soc. 135, 1741–1752 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Müller V.: On the Salas theorem and hypercyclicity of f(T). Integr. Equ. Oper. Theory 67(3), 439–448 (2010)zbMATHCrossRefGoogle Scholar
  16. 16.
    Protopopescu V., Azmy Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2(1), 79–90 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Rolewicz S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Salas H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 93–1004 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shapiro, J.H.: Simple connectivity and linear chaos. Rend. Circ. Mat. Palermo. (2) Suppl. 56, 27–48 (1998)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • J. Alberto Conejero
    • 1
  • Félix Martínez-Giménez
    • 1
    Email author
  1. 1.Departamento Matemática Aplicada, IUMPAUniversidad Politécnica de ValenciaValenciaSpain

Personalised recommendations