Archiv der Mathematik

, Volume 104, Issue 3, pp 223–235 | Cite as

On S. Grivaux’ example of a hypercyclic rank one perturbation of a unitary operator

Article

Abstract

Recently, Sophie Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. We give a similar construction using a functional model for rank one perturbations of singular unitary operators.

Keywords

Hypercyclic operator Rank one perturbation Inner function Model space Functional model 

Mathematics Subject Classification

47A16 30A76 30H10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Baranov and D. Yakubovich, Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators, arXiv:1212.5965.
  2. 2.
    Bayart F., Grivaux S.: Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358, 5083–5117 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge University Press, 2009.Google Scholar
  4. 4.
    K.C. Chan and J.H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), 1421–1449.Google Scholar
  5. 5.
    Clark D.N.: One-dimensional perturbations of restricted shifts. J. Anal. Math. 25, 169–191 (1972)CrossRefMATHGoogle Scholar
  6. 6.
    R.M. Gethner and J.H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281–288.Google Scholar
  7. 7.
    G. Godefroy and J.H. Shapiro, Operators with dense, invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229–269.Google Scholar
  8. 8.
    Grivaux S.: A new class of frequently hypercyclic operators. Indiana Univ. Math. J. 60, 1177–1201 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    S. Grivaux, A hypercyclic rank one perturbation of a unitary operator, Math. Nachr. 285 (2012), 533–544.Google Scholar
  10. 10.
    K.-G. Grosse-Erdmann A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011.Google Scholar
  11. 11.
    V.V. Kapustin, One-dimensional perturbations of singular unitary operators, Zapiski Nauchn. Sem. POMI 232 (1996), 118–122; English transl. in J. Math. Sci. (New York) 92 (1998), 3619–3621.Google Scholar
  12. 12.
    N.K. Nikol’skii, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.Google Scholar
  13. 13.
    N.K. Nikolski, Operators, Functions, and Systems: an Easy Reading, Math. Surveys Monogr., Vol. 92–93, AMS, Providence, RI, 2002.Google Scholar
  14. 14.
    A.G. Poltoratski, Boundary behavior of pseudocontinuable functions, Algebra i Analiz 5 (1993), 189–210; English transl. in St. Petersburg Math. J. 5 (1994), 389–406.Google Scholar
  15. 15.
    Shkarin S.: A hypercyclic finite rank perturbation of a unitary operator. Math. Ann. 348, 379–393 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.National Research University Higher School of EconomicsSt. PetersburgRussia
  3. 3.Chebyshev LaboratorySt. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations