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Multiplication Operators on the Lipschitz Space of a Tree

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Abstract

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space \({\mathcal{L}}\) of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on \({\mathcal{L}}\) that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on \({\mathcal{L}}\) are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of \({\mathcal{L}}\) we call the little Lipschitz space.

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References

  1. Allen R.F., Colonna F.: On the isometric composition operators on the Bloch space in ℂn. J. Math. Anal. Appl. 355(2), 675–688 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allen R.F., Colonna F.: Isometries and spectra of multiplication operators on the Bloch space. Bull. Aust. Math. Soc. 79, 147–160 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Allen R.F., Colonna F.: Multiplication operators on the Bloch space of bounded homogeneous domains. Comput. Methods Funct. Theory 9(2), 679–693 (2009)

    MATH  MathSciNet  Google Scholar 

  4. Allen R.F., Colonna F.: Weighted composition operators on the Bloch space of a bounded homogeneous domain. Oper. Theory Adv. Appl. 202, 11–37 (2009)

    Google Scholar 

  5. Allen R.F., Colonna F.: Weighted composition operators from H ∞ to the Bloch space of a bounded homogeneous domain. Integr. Equ. Oper. Theory 66, 21–40 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anderson J.M., Clunie J., Pommerenke Ch.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)

    MATH  MathSciNet  Google Scholar 

  7. Arazy, J.: Multipliers of Bloch Functions, vol. 54. University of Haifa Mathematics Publications, Israel (1982)

  8. Brown L., Shields A.L.: Multipliers and cyclic vectors in the Bloch space. Michigan Math. J. 38, 141–146 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cartier P.: Fonctions harmoniques sur un arbre. Symp. Math. 9, 203–270 (1972)

    Google Scholar 

  10. Cohen J.M., Colonna F.: The Bloch space of a homogeneous tree. Bol. Soc. Mat. Mexicana (2) 37(1-2), 63–82 (1992)

    MATH  MathSciNet  Google Scholar 

  11. Cohen J.M., Colonna F.: Embeddings of trees in the hyperbolic disk. Complex Variables 94, 311–335 (1994)

    MathSciNet  Google Scholar 

  12. Cohen J.M., Colonna F.: Extreme points of the Bloch space of a homogeneous tree. Israel J. Math. 94, 247–271 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Colonna F.: Bloch and normal functions and their relation. Rend. Circ. Mat. Palermo II XXXVIII, 161–180 (1989)

    Article  MathSciNet  Google Scholar 

  14. Conway J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (1990)

    MATH  Google Scholar 

  15. Duren P.L., Romberg B.W., Shields A.L.: Linear functions on H p spaces with 0 < p < 1. J. Reine Angew. Math. 238, 32–60 (1969)

    MATH  MathSciNet  Google Scholar 

  16. Hahn K.T.: Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem. Can. J. Math. 27, 446–458 (1975)

    MATH  Google Scholar 

  17. Hosokawa T., Izuchi K., Ohno S.: Topological structure of the space of weighted composition operators on H ∞. Integr. Equ. Oper. Theory 53, 509–526 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kolmogorov A.N., Fomin S.V.: Introductory Real Analysis. Dover, New York (1970)

    MATH  Google Scholar 

  19. Krantz S., Ma D.: Bloch functions on strongly pseudoconvex domains. Indiana Univ. Math. J. 37, 145–163 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li S.X.: Essential norm of the weighted composition operators on the Bloch spaces over the polydisk (Chinese, English and Chinese summary). Acta Math. Sinica (Chin. Ser.) 51(4), 655–662 (2008)

    MATH  MathSciNet  Google Scholar 

  21. Li, S., Stević, S.: Weighted composition operators from H ∞ to the Bloch space on the polydisc. Abstr. Appl. Anal. 2007, 13 (2007). (Article ID 48478)

  22. Li S., Stević S.: Weighted composition operators from α-Bloch spaces to H ∞ on the polydisk. Numer. Funct. Anal. Optim. 28(7), 911–925 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li S., Stević S.: Weighted composition operators between H ∞ and α-Bloch spaces in the unit ball. Taiwan. J. Math. 12, 1625–1639 (2008)

    MATH  Google Scholar 

  24. MacCluer B.D., Zhao R.: Essential norms of weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33(4), 1437–1458 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Ohno S.: Weighted composition operators between H ∞ and the Bloch space. Taiwan. J. Math. 5(3), 555–563 (2001)

    MATH  MathSciNet  Google Scholar 

  26. Ohno S., Zhao R.: Weighted composition operators on the Bloch space. Bull. Aust. Math. Soc. 63, 177–185 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Stević S.: Norm of weighted composition operators from Bloch space to H ∞ on the unit ball. Ars. Combin. 88, 125–127 (2008)

    MathSciNet  Google Scholar 

  28. Timoney R.M.: Bloch functions in several complex variables I. Bull. Lond. Math. Soc. 12, 241–267 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  29. Timoney R.M.: Bloch functions in several complex variables II. J. Reine Angew. Math. 319, 1–22 (1980)

    MATH  MathSciNet  Google Scholar 

  30. Weighted composition operators from H ∞ into α-Bloch spaces on the unit ball. Acta Math. Sinica (English) 25, 265–278 (2009)

    Google Scholar 

  31. Zhou Z., Chen R.: Weighted composition operators from F(p, q, s) to Bloch type spaces on the unit ball. Int. J. Math. 19, 899–926 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhu K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)

    MATH  Google Scholar 

  33. Zhu K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005)

    Google Scholar 

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Authors and Affiliations

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

    Flavia Colonna

  2. System Planning Corporation, Arlington, VA, 22201, USA

    Glenn R. Easley

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  1. Flavia Colonna
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  2. Glenn R. Easley
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Correspondence to Flavia Colonna.

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Colonna, F., Easley, G.R. Multiplication Operators on the Lipschitz Space of a Tree. Integr. Equ. Oper. Theory 68, 391–411 (2010). https://doi.org/10.1007/s00020-010-1824-5

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  • DOI: https://doi.org/10.1007/s00020-010-1824-5

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