Schatten class Toeplitz operators on Bergman spaces with almost standard weights

Document Type : Research Paper

Author

Faculty of Engineering, Ardakan University, P. O. 184, Ardakan, Iran

Abstract

We study Schatten class Toeplitz operators on weighted Bergman spaces induced by almost standard radial weights on the unit disk. We obtain a complete characterization of such operators generated by positive Borel measures. The characterization is given in terms of the Berezin transform, integrability of localized averages with respect to the M\"obius invariant measure, and discrete summability over pseudohyperbolic lattices. For Toeplitz operators generated by complex Borel measures, we establish sufficient conditions for Schatten class membership in terms of discrete lattice averages of the total variation, together with corresponding norm estimates. As an application, we derive Schatten class bounds for differences of such operators.

Keywords

Main Subjects

References

[1] Arazy, J., Fisher, S. D., & Peetre, J. (1988). Hankel operators on weighted Bergman spaces. Amer. J. Math., 110(6), 989{1053. https://doi.org/10.2307/2374685
[2] Arroussi, H., Park, I., & Pau, J. (2015). Schatten class Toeplitz operators acting on large weighted Bergman spaces. Studia Math., 229(3), 203{221. https://doi.org/10.4064/sm8345-12-2015
[3] Constantin, O. (2010). Carleson embeddings and some classes of operators on weighted Bergman spaces. J. Math. Anal. Appl., 365(2), 668{682. https://doi.org/10.1016/j.jmaa.2009.11.035
[4] Cuckovic, Z., & Zhao, R. (2004).Weighted composition operators on the Bergman space. J. London Math. Soc., 70(2), 499{511. https://doi.org/10.1112/S0024610704005605
[5] Duan, Y., Guo, K., Wang, S., & Wang, Z. (2022). Toeplitz operators on weighted Bergman spaces induced by a class of radial weights. J. Geom. Anal., 32, Paper No. 39,29 pp. https://doi.org/10.1007/s12220-021-00777-z
[6] Duren, P., & Schuster, A. (2004). Bergman Spaces. Mathematical Surveys and Monographs, 100. American Mathematical Society, Providence, RI. https://doi.org/10.1090/surv/100
[7] Esmaeili, K., & Kellay, K. (2023). Weighted composition operators on weighted Bergman and Dirichlet spaces. Canad. Math. Bull., 66(1), 286{302. https://doi.org/10.4153/S0008439522000297
[8] Esmaeili, K. (2026). A new characterization for the essential norm of weighted composition operators and their di erences on weighted Bergman spaces. Adv. Oper. Theory, 11(2), Paper No. 22, 27 pp. https://doi.org/10.1007/s43036-026-00497-7
[9] Huang, W., Huang, L., & Wang, X. (2024). Schatten class properties and essential norm estimates of operators on Bergman spaces induced by regular weights of annulus. Banach J. Math. Anal., 18(4), Paper No. 68, 48 pp. https://doi.org/10.1007/s43037-024-00378-2
[10] Lin, P., & Rochberg, R. (1996). Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights. Paci c J. Math., 173(1), 127{146.
[11] Luecking, D. H. (1987). Trace ideal criteria for Toeplitz operators. J. Funct. Anal., 73(2), 345{368. https://doi.org/10.1016/0022-1236(87)90072-3
[12] Pelaez, J. A., & Rattya, J. (2014). Weighted Bergman spaces induced by rapidly increasing weights. Memoirs Amer. Math. Soc., 227, No. 1066. https://doi.org/10.1090/memo/1066
[13] Pelaez, J. A., & Rattya, J. (2016). Trace class criteria for Toeplitz and composition operators on small Bergman spaces. Adv. Math., 293, 606{643. https://doi.org/10.1016/j.aim.2016.02.017
[14] Wang, X., Xia, J., & Liu, Y. (2023). Schatten class operators on exponential weighted Bergman spaces. J. Inequal. Appl., 2023, Paper No. 129, 26 pp. https://doi.org/10.1186/s13660-023-03031-y
[15] Yang, W., & Liu, J. (2023). Schatten classes of Toeplitz operators on Bergman{Besov Hilbert spaces in the unit ball. J. Math. Anal. Appl., 526(2), Paper No. 127257. https://doi.org/10.1016/j.jmaa.2023.127257
[16] Zeng, Z., Hu, Z., & Wang, X. (2025). Bounded, compact and Schatten class Hankel operators on Fock-type spaces. Trans. Amer. Math. Soc., 378(2), 805{849. https://doi.org/10.1090/tran/9290
[17] Zhu, K. (1988). Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Operator Theory, 20(2), 329{357.
[18] Zhu, K. (2007). Operator Theory in Function Spaces. Second edition. Mathematical Surveys and Monographs, 138. American Mathematical Society, Providence, RI. https://doi.org/10.1090/surv/138 

Files

History

  • Receive Date: 26 January 2026
  • Revise Date: 04 May 2026
  • Accept Date: 22 May 2026

Share

How to cite

Statistics