Advertisement

Hypercyclicity of Composition Operators on Discrete Weighted Banach Spaces

  • Robert F. Allen
  • Flavia Colonna
  • Rubén A. Martínez-AvendañoEmail author
  • Matthew A. Pons
Article
  • 47 Downloads

Abstract

In this paper, we study the hypercyclic composition operators on weighted Banach spaces of functions defined on discrete metric spaces. We show that the only such composition operators act on the “little” spaces. We characterize the bounded composition operators on the little spaces, as well as provide various necessary conditions for hypercyclicity.

Keywords

Composition operators Hypercyclicity weighted Banach spaces 

Mathematics Subject Classification

Primary 47B33 47A16 Secondary 47B38 

Notes

References

  1. 1.
    Allen, R.F., Colonna, F., Easley, G.R.: Composition operators on the Lipschitz space of a tree. Mediterr. J. Math. 11, 97–108 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Allen, R.F., Craig, I.M.: Multiplication operators on weighted Banach spaces of a tree. Bull. Korean Math. Soc. 54, 747–761 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Allen, R.F., Pons, M.A.: Composition operators on weighted Banach spaces of a tree. Bull. Malays. Math. Sci. Soc. (2016).  https://doi.org/10.1007/s40840-016-0428-x CrossRefzbMATHGoogle Scholar
  4. 4.
    Allen, RF., Pons, MA.: Weighted composition operators on discrete weighted Banach spaces, preprintGoogle Scholar
  5. 5.
    Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  6. 6.
    Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Mich. Math. J. 40, 271–297 (1993)CrossRefGoogle Scholar
  7. 7.
    Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127, 137–168 (1998)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bierstedt, K.D., Meise, R.G., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272, 107–160 (1982)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Colonna, F., Easley, G.R.: Multiplication operators on the Lipschitz space of a tree. Integr. Equ. Oper. Theory 68, 391–411 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Colonna, F., Martínez-Avendaño, R.A.: Hypercyclicity of composition operators on Banach spaces of analytic functions. Complex Anal. Oper. Theory 12, 305–323 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)zbMATHGoogle Scholar
  12. 12.
    Grosse-Erdmann, K.G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139, 47–68 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Grosse-Erdmann, K.G., Peris, A.: Linear Chaos. Springer, London (2011)CrossRefGoogle Scholar
  14. 14.
    Liang, Y., Zhou, Z.: Hypercyclic behaviour of multiples of composition operators on weighted Banach spaces of holomorphic functions. Bull. Belg. Math. Soc. Simon Stevin 21, 385–401 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lusky, W.: On the structure of \(H_{v_0}(D)\) and \(h_{v_0}(D)\). Math. Nachr. 159, 279–289 (1992)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Martínez-Avendaño, R.A.: Hypercyclicity of shifts on weighted \(L^p\) spaces of directed trees. J. Math. Anal. Appl. 446, 823–842 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Megginson, R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)CrossRefGoogle Scholar
  18. 18.
    Miralles, A., Wolf, E.: Hypercyclic composition operators on \(H^0_v\)-spaces. Math. Nachr. 286, 34–41 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Muthukumar, P., Ponnusamy, S.: Discrete analogue of generalized Hardy spaces and multiplication operators on homogenous trees. Anal. Math. Phys. (2017).  https://doi.org/10.1007/s13324-016-0141-9 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Muthukumar, P., Ponnusamy, S.: Composition operators on the discrete analogue of generalized Hardy space on homogenous trees. Bull. Malays. Math. Sci. Soc. (2017).  https://doi.org/10.1007/s40840-016-0419-y MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nordgren, E.A.: Composition operators. Can. J. Math. 20, 442–449 (1968)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Robert F. Allen
    • 1
  • Flavia Colonna
    • 2
  • Rubén A. Martínez-Avendaño
    • 3
    Email author
  • Matthew A. Pons
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of Wisconsin- La CrosseLa CrosseUSA
  2. 2.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  3. 3.Departamento Académico de MatemáticasInstituto Tecnológico Autónomo de MéxicoMexico CityMexico
  4. 4.Department of MathematicsNorth Central CollegeNapervilleUSA

Personalised recommendations