Abstract
In this paper, we study the bicomplex version of weighted Bergman spaces and the composition operators acting on them. We also investigate the Bergman kernel, duality properties and Berezin transform. This paper is essentially based on the work of Zhu (Operator Theory in Function Spaces of Math. Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence, 2007).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Not applicable.
References
Alpay, D., Luna-Elizarraras, M. E., Shapiro, M., Sruppa, D. C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Springer Briefs in Mathematics (2014)
Colombo, F., Sabadini, I., Struppa, D.C.: Bicomplex holomorphic functional calculus. Math. Nachr. 287(13), 1093–1105 (2013)
Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)
Gervais Lavoie, R., Marchildon, L., Rochon, D.: Infinite-dimensional bicomplex Hilbert spaces. Ann. Funct. Anal 1(2), 75–91 (2010)
Gervais Lavoie, R., Marchildon, L., Rochon, D.: Finite dimensional bicomplex Hilbert spaces. Adv. Appl. Clifford Algebras 21, 561–581 (2011)
Ghosh, C.: Bicomplex M\(\ddot{o}\)bius transformation (2018). arXiv:1706.07699
Hedenmalm, H., Koorenblum, B., Zhu, K.: Theory of Bergman Spaces. Graduate Text in Mathematics (2000)
Kumar, R., Singh, K.: Bicomplex linear operators on Hilbert spaces and Littlewood’s subordination principle. Adv. Appl. Clifford. Algebras 25, 591–560 (2015)
Kumar, R., Kumar, R., Rochon, D.: The fundamental theorems in the framework of bicomplex topological modules (2011). arXiv:1109.3424v1
Kumar, R., Singh, K., Saini, H., Kumar, S.: Bicomplex weighted Hardy spaces and bicomplex \(C^{*}\)- algebras. Adv. Appl. Clifford. Algebras 126, 217–235 (2016)
Luna-Elizarraras, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex numbers and their elementary functions. Cubo 14(2), 61–80 (2012)
Luna-Elizarraras, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex holomorphic functions: the algebra geometry and analysis of bicomplex numbers. Frontiers in Mathematics, Springer, New York (2015)
Reséndis O, L.F., Tovar, L.M.S.: Bicomplex Bergman and Bloch spaces. Arab. J. Math 9, 665–679 (2020)
Zhu, K.: Operator Theory in Function Spaces of Math. Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence (2007)
Acknowledgements
The authors would like to thank the referee for his\(\backslash \)her critical, helpful comments and valuable suggestions for improving this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fabrizio Colombo.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work of the second author is supported by NBHM (DAE) (Grant No. 2/11/41/2017/R &D-II/3480)
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
Dolkar, S., Kumar, S. Bicomplex Weighted Bergman Spaces and Composition Operators. Adv. Appl. Clifford Algebras 33, 46 (2023). https://doi.org/10.1007/s00006-023-01291-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-023-01291-x
Keywords
- Bicomplex holomorphic functions
- Bicomplex weighted Bergman spaces
- Composition operators
- Kernel functions