Advertisement

The Forward and Backward Shift on the Lipschitz Space of a Tree

  • Rubén A. Martínez-AvendañoEmail author
  • Emmanuel Rivera-Guasco
Article
  • 7 Downloads

Abstract

We initiate the study of the forward and backward shifts on the Lipschitz space of an undirected tree, \(\mathcal {L}\), and on the little Lipschitz space of an undirected tree, \(\mathcal {L}_0\). We determine that the forward shift is bounded both on \(\mathcal {L}\) and on \(\mathcal {L}_0\) and, when the tree is leafless, it is an isometry; we also calculate its spectrum. For the backward shift, we determine when it is bounded on \(\mathcal {L}\) and on \(\mathcal {L}_0\), we find the norm when the tree is homogeneous, we calculate the spectrum for the case when the tree is homogeneous, and we determine, for a general tree, when it is hypercyclic.

Keywords

Hypercyclicity Lipschitz space Trees Shifts 

Mathematics Subject Classification

47A16 47B37 05C05 05C63 

Notes

References

  1. 1.
    Allen, R.F., Colonna, F., Easley, G.R.: Multiplication operators between Lipschitz-type spaces on a tree. Int. J. Math. Math. Sci. Article ID 472495 (2011)Google Scholar
  2. 2.
    Allen, R.F., Colonna, F., Easley, G.R.: Multiplication operators on the iterated logarithmic Lipschitz spaces of a tree. Mediterr. J. Math. 9, 575–600 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Allen, R.F., Colonna, F., Easley, G.R.: Multiplication operators on the weighted Lipschitz space of a tree. J. Oper. Theory 69, 209–231 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  6. 6.
    Birkhoff, G.D.: Démonstration d’un théorème élémentaire sur les fonctions entières. C. R. Acad. Sci. Paris 189, 473–475 (1929)zbMATHGoogle Scholar
  7. 7.
    Cartier, P.: Fonctions harmoniques sur un arbre. In: Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971) Academic Press, London (1972)Google Scholar
  8. 8.
    Cartier, P.: Géométrie et analyse sur les arbres. In: Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 407. Lecture Notes in Mathematics, vol. 317, pp. 123–40. Springer, Berlin (1973)Google Scholar
  9. 9.
    Cohen, J.M., Colonna, F.: The Bloch space of a homogeneous tree. Bol. Soc. Mat. Mexicana 2(37), 63–82 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cohen, J.M., Colonna, F.: Embeddings of trees in the hyperbolic disk. Complex Var. Theory Appl. 24, 311–335 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Colonna, F., Easley, G.R.: Multiplication operators on the Lipschitz space of a tree. Integr. Equ. Oper. Theory 68, 391–411 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Colonna, F., Easley, G.R.: Multiplication operators between the Lipschitz space and the space of bounded functions on a tree. Mediterr. J. Math. 9, 423–438 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Colonna, F., Martínez-Avendaño, R.A.: Some classes of operators with symbol on the Lipschitz space of a tree. Mediterr. J. Math. 14(1), 18 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Colonna, F., Martínez-Avendaño R.A.: Composition operators on the little Lipschitz space of a tree. Preprint Google Scholar
  15. 15.
    Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, Berlin (1990)zbMATHGoogle Scholar
  16. 16.
    Fleming, R.J., Jamison, J.E.: Isometries on Banach spaces: function spaces. In: Monographs and Surveys in Pure and Applied Mathematics, 129. Chapman & Hall/CRC, Boca Raton (2003)Google Scholar
  17. 17.
    Grosse-Erdmann, K.G., Peris-Manguillot, A.: Linear Chaos. Springer, London (2006)zbMATHGoogle Scholar
  18. 18.
    Halmos, P.R.: A Hilbert Space Problem Book. Springer, Berlin (1982)CrossRefGoogle Scholar
  19. 19.
    Jabłoński, Z.J., Jung, I.B., Stochel, J.: Weighted shifts on directed trees. Mem. Am. Math. Soc. 216, 1017 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    MacLane, G.R.: Sequences of derivatives and normal families. J. Anal. Math. 2, 72–87 (1952) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Martínez-Avendaño, R.A.: Hypercyclicity of shifts on weighted \(\mathbf{L}^{p}\) spaces of directed trees. J. Math. Anal. Appl. 446, 823–842 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rolewicz, S.: On orbits of elements. Studia Math. 32, 17–22 (1969)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Shields, A.: Weighted shift operators and analytic function theory. In: Topics in Operator Theory, Mathematical Surveys, No. 13, pp. 49–128. American Mathematical Society, Providence, RI (1974)CrossRefGoogle Scholar
  24. 24.
    Zhu, K.: Operator Theory in Function Spaces, 2nd edn. American Mathematical Society, Providence, RI (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Rubén A. Martínez-Avendaño
    • 1
    Email author
  • Emmanuel Rivera-Guasco
    • 2
  1. 1.Departamento Académico de MatemáticasInstituto Tecnológico Autónomo de MéxicoMexico CityMexico
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMexico

Personalised recommendations