Advertisement

Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 2, pp 411–427 | Cite as

n-SOT Hypercyclic Linear Maps on Banach Algebra of Operators

  • Narjes Avizeh
  • Hamid RezaeiEmail author
Original Paper
  • 11 Downloads

Abstract

Let B(X) be the algebra of bounded linear operators on a Banach space X. A subset E of B(X) is said to be n-SOT dense in B(X) if for every continuous linear operator \(\Lambda \) from B(X) onto \(X^{(n)}\), the direct sum of n copies of X, \(\Lambda (E)\) is dense in \(X^{(n)}\). We consider the n-SOT hypercyclic continuous linear maps on B(X), namely, those that have orbits that are n-SOT dense in B(X). Some nontrivial examples of such operators are provided and many of their basic properties are investigated. In particular, we show that the left multiplication operator \(L_T\) is 1-SOT hypercyclic if and only if T is hypercyclic on X.

Keywords

Hypercyclic vector Hypercyclic operator Strong operator topology 

Mathematics Subject Classification

Primary 47A16 Secondary 47B48 

Notes

Acknowledgements

This paper is a part of the N. Avizeh’s doctoral thesis written at Yasouj University under the direction of the H. Rezaei.

References

  1. 1.
    Ansari, M., Hedayatian, K., Khani Robati, B., Moradi, A.: A note on topological and strict transitivity. Iran. J. Sci. Technol. Trans. Sci. 42(1), 59–64 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayart, F., Matheron, E.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bayart, F., Matheron, E.: Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250(2), 426–441 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bes, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Birkhoff, G.D.: D’emonstration d’un th’eoreme elementaire sur les fonctions enti’eres. C.R. Acad. Sci. Paris 189, 473–475 (1929)Google Scholar
  6. 6.
    Bonet, J., Martinez-Gimenez, F., Peris, A.: Universal and chaotic multipliers on spaces of operators. J. Math. Anal. Appl. 297, 599–611 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chan, K.C.: Hypercyclicity of the operator algebra for a separable Hilbert space. J. Oper. Theory 42, 231–244 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chan, K.C., Taylor, R.D.: Hypercyclic subspaces of a Banach space. Integral Equ. Oper. Theory 41, 381–388 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Feldman, N.S.: \(n\)-Weakly hypercyclic and \(n\)-weakly supercyclic operators. J. Funct. Anal. 263, 2255–2299 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grosse-Erdmann, K.G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. (N.S.) 36, 345–381 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grosse-Erdmann, K.G., Peris, A.: Linear Chaos. Universitext. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    MacLane, G.R.: Sequences of derivatives and normal families. J. d’Analyse Mathematique 2, 72–87 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Martinez-Gimenez, F., Perise, A.: Universality and chaos for tensor products of operators. J. Approx. Theory 124(1), 7–24 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Petersson, H.: A hypercyclicity criterion with applications. J. Math. Anal. Appl. 327, 1431–1443 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Petersson, H.: Hypercyclic conjugate operators. Integral Equ. Oper. Theory 57, 413–423 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rolewicz, S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shkarin, S.: Orbits of coanalytic Toeplitz operators and weak hypercyclicity. arXiv:1210.3191 [math.FA]
  19. 19.
    Shkarin, S.: Remarks on common hypercyclic vectors. J. Funct. Anal. 258, 132–160 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesYasouj UniversityYasoujIran

Personalised recommendations