Chaotic extensions of operators on Hilbert subspaces

Article

Abstract

If M is a closed subspace of a separable, infinite dimensional Hilbert space H with dim (H/M) = ∞, we show that every bounded linear operator A: MM can be extended to a chaotic operator T: HH that satisfies the hypercyclicity criterion in the strongest possible sense.

Keywords

Chaotic extension Hypercyclic extension Invariant subspace Infinite codimension 

Mathematics Subject Classification (2000)

47A16 46A22 47A15 

References

  1. 1.
    Argyros, S.A., Haydon, R.G.: A hereditarily indecomposable L space that solves the scalar-plus-compact problem. Preprint (2009)Google Scholar
  2. 2.
    Bayart F., Matheron E.: Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250, 426–441 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bès J.P., Chan K.C.: Approximation by chaotic operators and by conjugate classes. J. Math. Anal. Appl. 284, 206–212 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bonet J., Martínez-Giménez F., Peris A.: A Banach space which admits no chaotic operator. Bull. Lond. Math. Soc. 33, 196–198 (2001)CrossRefMATHGoogle Scholar
  5. 5.
    Bonet J., Martínez-Giménez F., Peris A.: Universal and chaotic multipliers on spaces of operators. J. Math. Anal. Appl. 297, 599–611 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chan K.C.: Hypercyclicity of the operator algebra for a separable space. J. Oper. Theory 42, 231–244 (1999)MATHGoogle Scholar
  7. 7.
    Chan K.C.: The density of hypercyclic operators on a Hilbert space. J. Oper. Theory 47, 131–143 (2001)Google Scholar
  8. 8.
    Chan K.C., Taylor R.D.: Hypercyclic subspace of a Banach space. Integral Equ. Oper. Theory 41, 381–388 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Conejero J.A., Peris A.: Linear transitivity criteria. Topol. Appl. 153, 767–773 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    De la Rosa M., Read C.: A hypercyclic operator whose direct sum \({T\oplus T}\) is not hypercyclic. J. Oper. Theory 61, 369–380 (2009)MathSciNetMATHGoogle Scholar
  11. 11.
    Feldman, N.: Linear chaos? http://home.wlu.edu/feldmann/research.html
  12. 12.
    Gethner G.M., Shapiro J.H.: Universal vectors for operators on spaces of holomorphic functions. Proc. Am. Math. Soc. 100, 281–288 (1987)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Godefroy G., Shapiro J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Grivaux S.: Topologically transitive extensions of bounded operators. Math. Z. 249, 85–96 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kitai, C.: Invariant closed sets for linear operators. PhD thesis, University of Toronto (1982)Google Scholar
  16. 16.
    León-Saavedra F., Müller V.: Hypercyclic sequences of operators. Stud. Math. 175, 1–18 (2006)CrossRefMATHGoogle Scholar
  17. 17.
    Montes-Rodríguez A.: Banach spaces of hypercyclic vectors. Mich. Math. J. 43, 419–436 (1996)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBowling Green State UniversityBowling GreenUSA

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