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Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

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Abstract

Let \({\varphi}\) be an analytic self-map of the unit disk \({\mathbb{D}}\), \({H(\mathbb{D})}\) the space of analytic functions on \({\mathbb{D}}\) and \({g \in H(\mathbb{D})}\). The boundedness and compactness of the operator \({DC_\varphi : H^\infty \rightarrow { \mathcal Z}}\) are investigated in this paper.

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References

  1. Choe B., Koo H., Smith W.: Composition operators on small spaces. Integral Equ. Oper. Theory 56, 357–380 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cowen C.C., MacCluer B.D.: Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

    Google Scholar 

  3. Čučković Ž., Zhao R.: Weighted composition operators between different weighted Bergman spaces and different Hardy spaces (English summary). Illinois J. Math. (electronic) 51(2), 479–498 (2007)

    MATH  Google Scholar 

  4. Duren P.L.: Theory of H p Spaces. Academic Press, New York (1970)

    MATH  Google Scholar 

  5. Hibschweiler R.A., Portnoy N.: Composition followed by differentiation between Bergman and Hardy spaces. Rocky Mountain J. Math. 35(3), 843–855 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Li S., Stević S.: Composition followed by differentiation between Bloch type spaces (English summary). J. Comput. Anal. Appl. 9(2), 195–205 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Li S., Stević S.: Composition followed by differentiation between weighted Bergman spaces and Bloch type spaces (English summary). J. Appl. Funct. Anal. 3(3), 333–340 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Li, S., Stević, S.: Volterra-type operators on Zygmund spaces. J. Inequal. Appl., vol. 2007, 10 p. (2007), Article ID 32124

  9. Li S., Stević S.: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 338, 1282–1295 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li S., Stević S.: Products of composition and integral type operators from H ∞ to the Bloch space. Complex Var. Elliptic Equ. 53(5), 463–474 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li S., Stević S.: Products of Volterra type operator and composition operator from H ∞ and Bloch spaces to Zygmund spaces. J. Math. Anal. Appl. 345, 40–52 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li S., Stević S.: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput. 206(2), 825–831 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, S., Stević, S.: Differentiation of a composition as an operator from spaces with mixed norm to Bloch α-spaces, Translated from the English (Russian. Russian summary) Mat. Sb., 199(12), 117–128 (2008); translation in Sb. Math., 199(11–12), 1847–1857 (2008)

  14. Li S., Stević S.: Cesàro type operators on some spaces of analytic functions on the unit ball. Appl. Math. Comput. 208, 378–388 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li S., Stević S.: Integral-type operators from Bloch-type spaces to Zygmund-type spaces. Appl. Math. Comput. 215, 464–473 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li S., Stević S.: Products of integral-type operators and composition operators between Bloch-type spaces. J. Math. Anal. Appl. 349, 596–610 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li S., Stević S.: Composition followed by differentiation between H ∞ and α-Bloch spaces (English summary). Houston J. Math. 35(1), 327–340 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Liu Y., Yu Y.: On a Li–Stević integral-type operators from the Bloch-type spaces into the logarithmic Bloch spaces. Integral Transforms Spec. Funct. 21(2), 93–103 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Madigan K., Matheson A.: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347(7), 2679–2687 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ohno S.: Products of composition and differentiation between Hardy spaces (English summary). Bull. Austral. Math. Soc. 73(2), 235–243 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Stević S.: Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces. Util. Math. 77, 167–172 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Stević, S.: On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball. Discrete Dyn. Nat. Soc., vol. 2008, 14 pp. (2008), Article ID 154263

  24. Stević S.: On a new operator from H ∞ to the Bloch-type space on the unit ball. Util. Math. 77, 257–263 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Stević S.: On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball. Appl. Math. Comput. 206, 313–320 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stević S.: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 354, 426–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stević S.: Integral-type operators from the mixed-norm space to the Bloch-type space on the unit ball. Siberian J. Math. 50(6), 1098–1105 (2009)

    Article  Google Scholar 

  28. Stević S.: Norm and essential norm of composition followed by differentiation from α-Bloch spaces to \({H^\infty_\mu}\). Appl. Math. Comput. 207, 225–229 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Stević S.: Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211, 222–233 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stević S.: Products of composition and differentiation operators on the weighted Bergman space. Bull. Belg. Math. Soc. Simon Stevin 16, 623–635 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Stević, S.: Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane. Abstr. Appl. Anal. vol. 2009, 8 pp. (2009), Article ID 161528

  32. Stević S.: On an integral operator from the Zygmund space to the Bloch-type space on the unit ball. Glasg. J. Math. 51, 275–287 (2009)

    Article  MATH  Google Scholar 

  33. Xiao J.: Holomorphic Q classes. Lecture Notes in Mathematics, vol. 1767. Springer, Berlin (2001)

    Google Scholar 

  34. Yang, W.: Products of composition and differentiation operators from Q K (p, q) spaces to Bloch-type spaces. Abstr. Appl. Anal. vol. 2009, 14 pp. (2009), Article ID 741920

  35. Yang W.: On an integral-type operator between Bloch-type spaces. Appl. Math. Comput. 215(3), 954–960 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  37. Zhu K.: Spaces of holomorphic functions in the unit ball. Graduate Text in Mathematics, vol. 226. Springer, New York (2005)

    Google Scholar 

  38. Zhu X.: Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces. J. Korean Math. Soc. 46(6), 1219–1232 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhu X.: Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces (English summary). Integral Transforms Spec. Funct. 18(3–4), 223–231 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhu X.: Extended Cesàro operator from H ∞ to Zygmund type spaces in the unit ball. J. Comput. Anal. Appl. 11(2), 356–363 (2009)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116, People’s Republic of China

    Yongmin Liu

  2. School of Mathematics and Physics Science, Xuzhou Institute of Technology, Xuzhou, 221008, People’s Republic of China

    Yanyan Yu

Authors
  1. Yongmin Liu
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  2. Yanyan Yu
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Corresponding author

Correspondence to Yongmin Liu.

Additional information

Communicated by Joseph Ball.

This work was completed with the support of the Natural Science Foundation of China (10471039) and the Grant of Higher Schools’ Natural Science Basic Research of Jiangsu Province of China (06KJD110175, 07KJB110115).

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Liu, Y., Yu, Y. Composition Followed by Differentiation Between H ∞ and Zygmund Spaces. Complex Anal. Oper. Theory 6, 121–137 (2012). https://doi.org/10.1007/s11785-010-0080-7

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