Abstract
Let \({\mathbb D}\) be the open unit disk in the complex plane. We characterize the boundedness and compactness of the sum of weighted differentiation composition operators
where \(n\in {\mathbb N}_0\), \(\psi _j\), \(j\in \overline{0,n}\), are holomorphic functions on \({\mathbb D}\), and \(\varphi \), a holomorphic self-maps of \({\mathbb D}\), acting from Bergman spaces with admissible weights to Zygmund type spaces.
Similar content being viewed by others
Data Availability
Not applicable.
References
Acharyya, S., Ferguson, T.: Sums of weighted differentiation composition operators. Complex Anal. Oper. Theory 13, 1465–1479 (2019)
Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)
Guo, Z., Liu, L., Shu, Y.: On Stević–Sharma operators from the mixed-norm spaces to Zygmund-type spaces. Math. Inequal. Appl. 24(2), 445–461 (2021)
Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman spaces. Springer, New York (2000)
Li, H., Guo, Z.: Note on a Li–Stević integral-type operator from mixed-norm spaces to \(n\)th weighted spaces. J. Math. Inequal. 11(1), 77–85 (2017)
Li, H., Li, S.: Norm of an integral operator on some analytic function spaces on the unit disk. J. Inequal. Appl. 2013, Article No. 342 (2013)
Li, S., Stević, S.: Composition followed by differentiation from mixed-norm spaces to \({\alpha }\)-Bloch spaces. Sb. Math. 199(12), 1847–1857 (2008)
Li, S., Stević, S.: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 338, 1282–1295 (2008)
Li, S., Stević, S.: Products of Volterra type operator and composition operator from \(H^\infty \) and Bloch spaces to the Zygmund space. J. Math. Anal. Appl. 345, 40–52 (2008)
Madigan, K., Matheson, A.: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347(7), 2679–2687 (1995)
Manhas, J., Zhao, R.: Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces. Acta Math. Sci. Ser. B (Engl. Ed.), 38, 1105–1120 (2018)
Sharma, A.K.: Products of multiplication, composition and differentiation between weighted Bergman–Nevanlinna and Bloch-type spaces. Turk. J. Math. 35, 275–291 (2011)
Sharma, A.K., Ueki, S.: Composition operators from Nevanlinna type spaces to Bloch type spaces. Banach J. Math. Anal. 6(2), 112–123 (2012)
Sharma, A.K., Kumari, R.: Weighted composition operators between Bergman and Bloch spaces. Commun. Korean Math. Soc. 22(3), 373–382 (2007)
Sharma, A.K., Sharma, S.D.: Weighted composition operators between Bergman-type spaces. Commun. Korean Math. Soc. 21(3), 465–474 (2006)
Sharma, A.K., Ueki, S.: Composition operators between weighted Bergman spaces with admissible Békollé weights. Banach J. Math. Anal. 8(1), 64–88 (2014)
Sharma, A.K., Sharma, M., Mursaleen, M.: Semigroups of composition operators on vector-valued Hardy spaces. Numer. Funct. Anal. Optim. 43(7), 876–885 (2022)
Sharma, A.K., Iqbal, J., Farooq, S., Mursaleen, M.: Weighted composition operators on spaces generated by fractional Cauchy kernels of the unit ball. J. Anal. (2024). https://doi.org/10.1007/s41478-023-00700-5
Stević, S.: Norm of weighted composition operators from \({\alpha }\)-Bloch spaces to weighted-type spaces. Appl. Math. Comput. 215, 818–820 (2009)
Stević, S.: Norm of some operators from logarithmic Bloch-type spaces to weighted-type spaces. Appl. Math. Comput. 218, 11163–11170 (2012)
Stević, S.: On a new product-type operator on the unit ball. J. Math. Inequal. 16(4), 1675–1692 (2022)
Stević, S.: Note on a new class of operators between some spaces of holomorphic functions. AIMS Math. 8(2), 4153–4167 (2023)
Stević, S., Huang, C.S., Jiang, Z.J.: Sum of some product-type operators from Hardy spaces to weighted-type spaces on the unit ball. Math. Methods Appl. Sci. 45(17), 11581–11600 (2022)
Stević, S., Sharma, A.K.: Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk. Appl. Math. Comput. 217, 10187–10194 (2011)
Stević, S., Sharma, A.K.: Essential norm of composition operators between weighted Hardy spaces. Appl. Math. Comput. 217, 6192–6197 (2011)
Stević, S., Sharma, A.K., Bhat, A.: Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces. Appl. Math. Comput. 218, 2386–2397 (2011)
Stević, S., Sharma, A.K., Bhat, A.: Products of multiplication composition and differentiation operators on weighted Bergman spaces. Appl. Math. Comput. 217, 8115–8125 (2011)
Stević, S., Sharma, A.K., Krishan, R.: Boundedness and compactness of a new product-type operator from a general space to Bloch-type spaces. J. Inequal. Appl. 2016, 219 (2016)
Stević, S., Ueki, S.I.: Polynomial differentiation composition operators from \(H^p\) spaces to weighted-type spaces on the unit ball. J. Math. Inequal. 17(1), 365–379 (2023)
Wang, S., Wang, M., Guo, X.: Products of composition, multiplication and iterated differentiation operators between Banach spaces of holomorphic functions. Taiwan. J. Math. 24(2), 355–376 (2020)
Acknowledgements
The authors are thankful to the referee for several helpful comments and suggestions.
Funding
Ajay K. Sharma is thankful to DST for the research project CRG/2023/002767 under (CRG)SERB scheme.
Author information
Authors and Affiliations
Contributions
All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Jesus Araujo Gomez.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sharma, A.K., Kumar, S., Sharma, M. et al. On sum of weighted differentiation composition operators from Bergman spaces with admissible weights to Zygmund type spaces. Adv. Oper. Theory 9, 45 (2024). https://doi.org/10.1007/s43036-024-00345-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43036-024-00345-6
Keywords
- Sum of weighted differentiation composition operator
- Bergman space with admissible weight
- Zygmund type space