Chaotic differential operators

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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 
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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)MATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHGoogle 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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHGoogle Scholar
  9. 9.
    Grosse-Erdmann K.-G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139(1), 47–68 (2000)MathSciNetMATHGoogle 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)MathSciNetMATHGoogle 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)MATHCrossRefGoogle Scholar
  14. 14.
    Martínez-Giménez F.: Chaos for power series of backward shift operators. Proc. Am. Math. Soc. 135, 1741–1752 (2007)MATHCrossRefGoogle Scholar
  15. 15.
    Müller V.: On the Salas theorem and hypercyclicity of f(T). Integr. Equ. Oper. Theory 67(3), 439–448 (2010)MATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Rolewicz S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)MathSciNetMATHGoogle 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
  1. 1.Departamento Matemática Aplicada, IUMPAUniversidad Politécnica de ValenciaValenciaSpain

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