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Complex Analysis and Operator Theory

, Volume 13, Issue 2, pp 479–492 | Cite as

Bloch Space of a Bounded Symmetric Domain and Composition Operators

  • Cho-Ho Chu
  • Hidetaka HamadaEmail author
  • Tatsuhiro Honda
  • Gabriela Kohr
Article
  • 123 Downloads

Abstract

We generalize a number of finite dimensional results on Bloch functions to infinite dimensional bounded symmetric domains. In particular, we characterize the Bloch space as well as the little Bloch space of a Hilbert ball, and give one sufficient and several necessary conditions for a composition operator on a Bloch space to be an isometry. We also answer some open questions of Allen and Colonna concerning Bloch functions and composition operators.

Keywords

Bloch space Bounded symmetric domain Composition operator JB\(^*\)-triple 

Mathematics Subject Classification

47B38 32A18 32M15 

References

  1. 1.
    Allen, R.F., Colonna, F.: On the isometric composition operators on the Bloch space in \({\mathbb{C}}^n\). J. Math. Anal. Appl. 355, 675–688 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blasco, O., Galindo, P., Miralles, A.: Bloch functions on the unit ball of an infinite dimensional Hilbert space. J. Funct. Anal. 267, 1188–1204 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chu, C.-H.: Jordan structures in geometry and analysis. In: Cambridge Tracts in Mathematics, vol. 190. Cambridge University Press, Cambridge (2012)Google Scholar
  4. 4.
    Chu, C.-H., Hamada, H., Honda, T., Kohr, G.: Distortion of locally biholomorphic Bloch mappings on bounded symmetric domains. J. Math. Anal. Appl. 441, 830–843 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chu, C.-H., Hamada, H., Honda, T., Kohr, G.: Bloch functions on bounded symmetric domains. J. Funct. Anal. 272, 2412–2441 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deng, F., Ouyang, C.: Bloch spaces on bounded symmetric domains in complex Banach spaces. Sci. China Ser. A Math. 49, 1625–1632 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Franzoni, T., Vesentini, E.: Holomorphic maps and invariant distances. In: Notas de Matemática [Mathematical Notes], vol. 69. North-Holland Publishing Co., Amsterdam (1980)Google Scholar
  8. 8.
    Hahn, K.T.: Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem. Canad. J. Math. 27, 446–458 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hamada, H.: Weighted composition operators from \(H^{\infty }\) to the Bloch space of infinite dimensional bounded symmetric domains. Complex Anal. Oper. Theory 12, 207–216 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hamada, H.: A distortion theorem and the Bloch constant for Bloch mappings in \({\mathbb{C}}^n\). J. Anal. Math. (to appear) Google Scholar
  11. 11.
    Hamada, H., Honda, T., Kohr, G.: Trace-order and a distortion theorem for linearly invariant families on the unit ball of a finite dimensional JB\(^*\)-triple. J. Math. Anal. Appl. 396, 829–843 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hamada, H., Honda, T., Kohr, G.: Growth and distortion theorems for linearly invariant families on homogeneous unit balls in \(\mathbb{C}^n\). J. Math. Anal. Appl. 407, 398–412 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hamada, H., Kohr, G.: Pluriharmonic mappings in \(\mathbb{C}^n\) and complex Banach spaces. J. Math. Anal. Appl. 426, 635–658 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hamada, H., Kohr, G.: \(\alpha \)-Bloch mappings on bounded symmetric domains in \(\mathbb{C}^n\). Complex Anal. Oper. Theory 12, 509–527 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Holland, F., Walsh, D.: Criteria for membership of Bloch space and its subspace, BMOA. Math. Ann. 273, 317–335 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183, 503–529 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Krantz, S.G., Ma, D.: Bloch functions on strongly pseudoconvex domains. Indiana Univ. Math. J. 37, 145–163 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Loos, O.: Bounded Symmetric Domains and Jordan Pairs. University of California, Irvine (1977)Google Scholar
  19. 19.
    Ren, G., Tu, C.: Bloch space in the unit ball of \({\mathbb{C}}^n\). Proc. Am. Math. Soc. 133, 719–726 (2005)CrossRefzbMATHGoogle Scholar
  20. 20.
    Stroethoff, K.: The Bloch space and Besov spaces of analytic functions. Bull. Aust. Math. Soc. 54, 211–219 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Timoney, R.M.: Bloch functions in several complex variables, I. Bull. Lond. Math. Soc. 12, 241–267 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Timoney, R.M.: Bloch functions in several complex variables, II. J. Reine Angew. Math. 319, 1–22 (1980)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Wicker, F.D.: Generalized Bloch mappings in complex Hilbert space. Can. J. Math. 29, 299–306 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Xiong, C.: Norm of composition operators on the Bloch space. Bull. Aust. Math. Soc. 70, 293–299 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhu, K.: Operator Theory in Function Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 139. Marcel Dekker, Inc., New York (1990)Google Scholar
  26. 26.
    Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, vol. 226. Springer, New York (2005)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Cho-Ho Chu
    • 1
  • Hidetaka Hamada
    • 2
    Email author
  • Tatsuhiro Honda
    • 3
    • 4
  • Gabriela Kohr
    • 5
  1. 1.Queen Mary, University of LondonLondonEngland
  2. 2.Faculty of Science and EngineeringKyushu Sangyo UniversityFukuokaJapan
  3. 3.Hiroshima Institute of TechnologyHiroshimaJapan
  4. 4.School of CommerceSenshu UniversityKawasakiJapan
  5. 5.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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