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Some classes of operators with symbol on the Lipschitz space of a tree

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Abstract

In this paper, we expand the study of the multiplication operators on the Lipschitz space of a tree begun in Colonna and Easley (Integral Equ Oper Theory 68:391–411, 2010) by focusing on their adjoint acting on a certain separable subspace of the Lipschitz space whose dual is isometrically isomorphic to \(\mathbf L^1\). We then study the properties of two useful operators \(\nabla \) and \(\Delta \) and use them (along with the multiplicative symbol \(\psi \)) to define the Toeplitz operator \(T_\psi \) on the space \(\mathbf L^p\) for \(1\le p \le \infty \). We give conditions for its boundedness and study its point spectrum.

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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allen, R.F., Colonna, F., Easley, G.R.: Multiplication operators between Lipschitz-type spaces on a tree. Int. J. Math. Math. Sci. 36 (2011) (Art. ID 472495)

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Allen, R.F., Craig, I.M.: Multiplication operators on weighted Banach spaces of a tree. Bull. Korean Math. Soc. (to appear)

  6. Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  7. Birkhoff, G.D.: Démonstration d’un théorème élémentaire sur les fonctions entières. C. R. Acad. Sci. Paris 189, 473–475 (1929)

    MATH  Google Scholar 

  8. Cartier, P.: Fonctions harmoniques sur un arbre. In: Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971) Academic Press, London (1972)

  9. Cartier, P.: Géométrie et analyse sur les arbres. In: Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 407, pp. 123–40. Lecture Notes in Math., Vol. 317. Springer, Berlin (1973)

  10. Cohen, J.M., Colonna, F.: Embeddings of trees in the hyperbolic disk. Complex Var. Theory Appl. 24, 311–335 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Colonna, F., Easley, G.R.: Multiplication operators on the Lipschitz space of a tree. Integral Equ. Oper. Theory 68, 391–411 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Godefroy, G., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grosse-Erdmann, K.-G., Manguillot, A.P.: Linear Chaos. Springer, London (2006)

    MATH  Google Scholar 

  15. Jabloński, Z.J., Jung, Il B., Stochel, J.: Weighted Shifts on Directed Trees. Mem. Am. Math. Soc., vol. 216, no. 1017, viii+106 pp. 106 (2012)

  16. MacLane, G.R.: Sequences of derivatives and normal families. J. Anal. Math. 2, 72–87 (1952/53)

  17. Martínez-Avendaño, R.A.: Hypercyclicity of shifts on weighted \({ L}^p\) spaces of directed trees. J. Math. Anal. Appl. 446, 823–842 (2017)

  18. Rolewicz, S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)

    MathSciNet  MATH  Google Scholar 

  19. Salas, H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347, 993–1004 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)

    Book  MATH  Google Scholar 

  21. Shields, A.: Weighted shift operators and analytic function theory. In: Pearcy, C.(ed.) Topics in Operator Theory, pp. 49–128. Math. Surveys, No. 13, American Mathematical Society, Providence (1974)

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Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Flavia Colonna

  2. Centro de Investigación en Matemáticas, Universidad Autónoma del Estado de Hidalgo, Pachuca, Hidalgo, Mexico

    Rubén A. Martínez-Avendaño

Authors
  1. Flavia Colonna
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  2. Rubén A. Martínez-Avendaño
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Corresponding author

Correspondence to Rubén A. Martínez-Avendaño.

Additional information

The second author wishes to thank the Department of Mathematical Sciences at George Mason University for their hospitality during the time of this research. The work of the second author was made possible by the “Programa de Estancias Sabáticas en el Extranjero 2015” of the Consejo Nacional de Ciencia y Tecnología, México. Both authors thank the referee for valuable comments and corrections.

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Colonna, F., Martínez-Avendaño, R.A. Some classes of operators with symbol on the Lipschitz space of a tree. Mediterr. J. Math. 14, 18 (2017). https://doi.org/10.1007/s00009-016-0805-6

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  • DOI: https://doi.org/10.1007/s00009-016-0805-6

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