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Weighted Composition Operators from H ∞ to the Bloch Space of a Bounded Homogeneous Domain

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Abstract

Let D be a bounded homogeneous domain in ℂn. In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space H ∞(D) into the Bloch space of D. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if D is a bounded symmetric domain, the bounded multiplication operators from H ∞(D) to the Bloch space of D are the operators whose symbol is bounded.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  4. R. F. Allen and F. Colonna, Weighted composition operators on the Bloch space of a bounded homogeneous domain, Proc. IWOTA (to appear).

  5. Cartan E.: Sur les domains bournés de l’espace de n variable complexes (French). Abh. Math. Sem. Univ. Hamburg 11, 116–162 (1935)

    Article  MATH  Google Scholar 

  6. R. Chen, S. Stević and Z. Zhou, Weighted composition operators between Bloch type spaces in the polydisk, Mat. Sb. (to appear).

  7. D. Clahane, S. Stević and Z. Zhou, On composition operators between Bloch type spaces in polydisc, arXiv:math\0507339v1.

  8. Cohen J.M., Colonna F.: Bounded holomorphic functions on bounded symmetric domains. Trans. Amer. Math. Soc. 343, 135–156 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cohen J.M., Colonna F.: Isometric composition operators on the Bloch space in the polydisk. Contemp. Math., 454, 9–21 (2008)

    MathSciNet  Google Scholar 

  10. Colonna F.: Characterisation of the isometric composition operators on the Bloch space. Bull. Austral. Math. Soc. 72, 283–290 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Drucker D.: Exception Lie algebras and the structure of Hermitian symmetric spaces. Mem. Amer. Math. Soc. 208, 1–207 (1978)

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  13. Helgason S.: Differential Geometry and Symmetric Spaces. Academic Press, New York (1962)

    MATH  Google Scholar 

  14. 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 

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

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

    Article  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  18. Ohno S.: Weighted composition operators between H ∞ and the Bloch space. Taiwanese J. Math. 5(3), 555563 (2001)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Pjateckiĭ-Šapiro I.I.: On a problem proposed by E. Cartan (Russian). Dokl. Akad. Nauk SSSR 124, 272–273 (1959)

    MathSciNet  Google Scholar 

  21. Stević S.: Composition operators between H ∞ and theα-Bloch spaces on the polydisc. Z. Anal. Anwendungen 25(4), 457–466 (2006)

    MATH  Google Scholar 

  22. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  25. Zhang G.: Bloch constants of bounded symmetric domains. Trans. Amer. Math. Soc. 349, 2941–2949 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang M., Chen H.: Weighted composition operators of H ∞ into α-Bloch spaces on the unit ball. Acta Math. Sinica (English) 25, 265–278 (2009)

    Article  MATH  Google Scholar 

  27. Z. Zhou and R. Chen, Weighted composition operators from F(p, q, s) to Bloch type spaces on the unit ball (preprint) (http://arxiv.org/abs/math/0503614v9).

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin–La Crosse, La Crosse, WI, 54601, USA

    Robert F. Allen

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

    Flavia Colonna

Authors
  1. Robert F. Allen
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  2. Flavia Colonna
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Correspondence to Robert F. Allen.

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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). https://doi.org/10.1007/s00020-009-1736-4

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  • DOI: https://doi.org/10.1007/s00020-009-1736-4

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