Abstract
We study composition operators between the weighted Bergman space of the polydisk and the weighted Bergman space of the unit disk. We find conditions on the symbols to guarantee the boundedness and compactness of the composition operator. We shall also find a necessary and sufficient condition for the composition operator to be Hilbert–Schmidt.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
datavailability
No new data were created or analyzed during this study. Data sharing is not applicable to this article.
References
Abkar, A.: Commutant of multiplication operators in weighted Bergman spaces on polydisk. Czech. Math. J. 70, 727–741 (2020)
Babaei, A., Abkar, A.: Weighted composition-differentiation operators on the Hardy and Bergman spaces. Bull. Iran. Math. Soc. 48, 3637–3658 (2022)
Conway, J.B.: A course in operator theory. American Mathematical Society. (2000)
Cowen, C.C., MacCluer, B.D.: Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press (1995)
Duren, P., Schuster, A.P.: Bergman spaces, Mathematical Surveys and Monographs. Vol. 100, American Mathematical Society, (2004)
Fatehi, M., Hammond, C.N.B.: Composition-differentiation operator on the Hardy space. Proc. Am. Math. Soc. 148(7), 2893–2900 (2020)
Fatehi, M., Hammond, C.N.B.: Norms of weighted composition operators with automorphic symbol. Integr. Equ. Oper. Theory. 92, no. 13, 12 pages (2020)
Izuchi, K.J., Nguyen, Q.D., Ohno, S.: Composition operators induced by analytic maps to the polydisk. Canadian J. Math. 64, 1329–1340 (2012)
Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman spaces, Graduate Texts in Mathematics. 199, Springer, (2000)
Liang, Y.X., Zhou, Z.H.: The products of differentiation and composition operators from logarithmic Bloch spaces to \(\mu \)-Bloch spaces. Bull. Iran. Math. Soc. 46, 159–176 (2020)
Mehrangiz, S., Khani-Robati, B.: On the convergence of sequences of generalized weighted composition operators on the Bloch type spaces. Iranian J. Sci. Tech. Transactions A: Science. 1-8 (2021)
Ohno, S.: Products of composition and differentiation between Hardy spaces. Bull. Austr. Math. Soc. 73(2), 235–243 (2006)
Stessin, M., Zhu, K.: Composition operators induced by symbols defend on a polydisk. J. Math. Anal. Appl. 319, 815–829 (2006)
Stessin, M., Zhu, K.: Composition operators on embedded disks. J. Operator Theory 56, 423–449 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fereshteh Sady.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Saeidikia, Z., Abkar, A. Composition Operators on Weighted Bergman Spaces of Polydisk. Bull. Iran. Math. Soc. 49, 31 (2023). https://doi.org/10.1007/s41980-023-00775-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41980-023-00775-7