Abstract
In the setting of Fock–Sobolev spaces of positive orders over the complex plane, Choe and Yang showed that if the one of the symbols of two commuting Toeplitz operators with bounded symbols is non-trivially radial, then the other must also be radial. In this paper, we extend this result to the Fock–Sobolev space of negative order using the Fock-type space with a confluent hypergeometric function.
This is a preview of subscription content, log in via an institution to check access.
References
Axler, S., and Čučkovic̀, Ž.: Commuting Toeplitz operators with harmonic symbols. Integral Equations Operator Theory, 14, 1–11 (1991)
Bateman, H.: Tables of Integral Transforms Volume I, McGraw-Hill, New York, 1954
Bauer, W.: Berezin Toeplitz quantization and composition formulas. J. Funct. Anal., 256, 3107–3142 (2009)
Bauer, W., Lee, Y. J.: Commuting Toeplitz operators on the Segal–Bargmann space. J. Funct. Anal., 260, 460–489 (2011)
Cho, H. R., Choe, B. R., Koo, H.: Fock–Sobolev spaces of fractional order. Potential Anal., 43, 199–240 (2015)
Cho, H. R., Choi, H., Lee, H.-W.: Explicit formula for the reproducing kernels of some weighted fock spaces. J. Math. Anal. Appl., 497(2), 124920 (2021)
Cho, H. R., Park, S.: Fractional Fock–Sobolev spaces. Nagoya Math. J., 237, 79–97 (2020)
Choe, B. R., Koo, H., Lee, Y. J.: Commuting Toeplitz operators on the polydisk. Trans. Amer. Math. Soc., 356, 1727–1749 (2004)
Choe, B. R., Lee, Y. J.: Pluriharmonic symbols of commuting Toeplitz operators. Illinois J. Math., 37, 424–436 (1993)
Choe, B. R., Yang, J.: Commutants of Toeplitz operators with radial symbols on the Fock–Sobolev space. J. Math. Anal. Appl., 415, 779–790 (2014)
Čučkovic̀, Ž., Rao, N. V.: Mellin Transform, Monomial Symbols, and Commuting Toeplitz Operators. J. Funct. Anal., 154, 195–214 (1998)
Dixit, A., Moll, V. H.: The integrals in Gradshteyn and Ryzhik. Part 28: The confluent hypergeometric function and Whittaker functions. Scientia, Series A: Mathematical Sciences, 26, 49–62 (2015)
Gradshteyn, I. S., Ryzhik, I. M.: Table of Integrals, Series and Products 5th edition, Academic Press, New York, 1993
Grudsky, S., Quiroga-Barranco, R., Vasilevski, N.: Commutative C*-algebras of Toeplitz operators and quantization on the unit disc. J. Funct. Anal., 234, 1–44 (2006)
Le, T.: The commutants of certain Toeplitz operators on weighted Bergman spaces. J. Math. Anal. Appl., 348, 1–11 (2008)
Lebedev, N. N.: Special Functions and Their Applications, Dover Publications, Inc., New York, 1972
Lee, Y. J.: Commuting Toeplitz operators on the Hardy space of the polydisk. Proc. Amer. Math. Soc., 138, 189–197 (2010)
Vasilevski, N.: Commutative Algebras of Toeplitz Operators on the Bergman Space. Oper. Theory Adv. Appl., Vol. 185, Birkhäuser, Basel, 2008
Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2004
Zhu, K.: Analysis on Fock spaces, Graduate Texts in Mathematics, Vol. 263, Springer, New York, 2012
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflict of interest.
Additional information
The first author was supported by NRF of Korea (Grant No. NRF-2020R1F1A1A01048601); the second author was supported by NRF of Korea (Grant No. NRF-2020R1I1A1A01074837)
Rights and permissions
About this article
Cite this article
Cho, H.R., Lee, HW. Commuting Toeplitz Operators on Fock–Sobolev Spaces of Negative Orders. Acta. Math. Sin.-English Ser. 39, 1989–2005 (2023). https://doi.org/10.1007/s10114-023-1541-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-023-1541-z
Keywords
- Fock–Sobolev spaces
- commutator of Toeplitz operators
- Mellin Transform
- Confluent Hypergeometric Function