Skip to main content
SpringerLink
Log in

New characterizations for differences of weighted differentiation composition operators from a Bloch-type space to a weighted-type space

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We found several new equivalent characterizations of the boundedness of the differences of weighted differentiation composition operators from Bloch-type spaces to weighted-type spaces. Especially, we estimated its essential norm in terms of the n-th power of the induced analytic self-maps on the unit disk, which can provide a new and simple compactness criterion. Moreover, we applied our results to a classical example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, FL, 1995)

    MATH  Google Scholar 

  2. J.N. Dai, C.H. Ouyang, Differences of weighted composition operators on \(H_\alpha ^\infty (B_N)\), J. Inequal. Appl. Article ID 127431 (2009), 19 pp

  3. P. Gorkin, B.D. MacCluer, Essential norms of composition operators. Integral Equ. Oper. Theory 48, 27–40 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. O. Hyvärinen, M. Lindström, Estimates of essential norms of weighted composition operators between Bloch-type spaces. J. Math. Anal. Appl. 393, 38–44 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. O. Hyvärinen, M. Kemppainen, M. Lindström, A. Rautio, E. Saukko, The essential norm of weighted composition operators on weighted Banach spaces of analytic functions. Integral Equ. Oper. Theory 72, 151–157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. X. Liu, S. Li, Differences of generalized weighted composition operators from the Bloch space into Bers-type spaces. Filomat 31(6), 1671–1680 (2017)

    Article  MathSciNet  Google Scholar 

  7. S. Li, S. Stević, Generalized weighted composition operators from \(\alpha \)-Bloch spaces into weighted-type spaces. J. Inequal. Appl. 2015, 265 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Y.X. Liang, X.T. Dong, New characterizations for the products of differentiation and composition operators between Bloch-type spaces. J. Inequal. Appl. 2014, 502 (2014). https://doi.org/10.1186/1029-242X-2014-502

    Article  MathSciNet  MATH  Google Scholar 

  9. Y.X. Liang, Z.H. Zhou, Essential norm of product of differentiation and composition operators between Bloch-type spaces. Arch. Math. 100(4), 347–360 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Y.X. Liang, Z.H. Zhou, New estimate of essential norm of composition followed by differentiation between Bloch-type spaces. Banach J. Math. Anal. 8, 118–137 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y.X. Liang, Z.H. Zhou, Weighted differentiation composition operator from logarithmic Bloch spaces to Bloch-type spaces. Math. Nachr. 290(2–3), 123–148 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Y.X. Liang, Z.H. Zhou, Differences of integral-type operators from weighted Bergman spaces to \(\beta \)-Bloch–Orlicz spaces. Publ. Math. Debrecen (2018) (preprint)

  13. M. Lindström, E. Wolf, Essential norm of the difference of weighted composition operators. Monatsh. Math. 153(2), 133–143 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. J.S. Manhas, R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces. J. Math. Anal. Appl. 389, 32–47 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Shi, S. Li, Essential norm estimates for differences of composition operators on the Bloch space. Math. Ineq. Appl. 20, 543–555 (2017)

    MATH  Google Scholar 

  17. Y. Shi, S. Li, Differences of composition operators on Bloch type spaces. Complex Anal. Oper. Theory 11(1), 227–242 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Stević, Characterizations of composition followed by differentiation between Bloch-type spaces. Appl. Math. Comput. 218, 4312–4316 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Y. Wu, H. Wulan, Products of differentiation and composition operators on the Bloch space. Collect. Math. 63, 93–107 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Wulan, D. Zheng, K. Zhu, Compact composition operators on \(BMOA\) and the Bloch space. Proc. Am. Math. Soc. 137, 3861–3868 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Zhao, Essential norms of composition operators between Bloch type spaces. Proc. Am. Math. Soc. 138, 2537–2546 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. X. Zhu, Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces. Indian J. Math. 49(2), 1219–1232 (2009)

    MathSciNet  MATH  Google Scholar 

  25. X. Zhu, Generalized weighted composition operators from Bloch spaces into Bers-type spaces. Math. Inequal. Appl. 17(1), 1163–1169 (2014)

    MATH  Google Scholar 

  26. X. Zhu, A new characterization of the generalized weighted composition operator from \(H^\infty \) into the Zygmund space. Math. Inequal. Appl. 18(3), 1135–1142 (2015)

    MathSciNet  MATH  Google Scholar 

  27. X. Zhu, Essential norm of generalized weighted composition operators on Bloch-type spaces. Appl. Math. Comput. 274, 133–142 (2016)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referee for useful comments and suggestions which improved the presentation of this paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, People’s Republic of China

    Yu-Xia Liang

Authors
  1. Yu-Xia Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu-Xia Liang.

Additional information

This paper was supported in part by the National Natural Science Foundation of China (Grant No. 11701422) and the Foundation of Tianjin Education Commission (Grant No. 2017KJ124).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, YX. New characterizations for differences of weighted differentiation composition operators from a Bloch-type space to a weighted-type space. Period Math Hung 77, 119–138 (2018). https://doi.org/10.1007/s10998-018-0239-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-018-0239-y

Keywords

Mathematics Subject Classification

Search

Navigation