Algebraic and spectral properties of H-Toeplitz operators on the Bergman space

Jie Zhang; Jie Zhang

doi:10.3934/math.2025603

AIMS Mathematics

2025, Volume 10, Issue 6: 13432-13450. doi: 10.3934/math.2025603

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Research article

Algebraic and spectral properties of H-Toeplitz operators on the Bergman space

Jie Zhang ^,

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received: 23 March 2025 Revised: 01 June 2025 Accepted: 05 June 2025 Published: 11 June 2025
MSC : 47B35

In this paper, we discuss the zero product problem of two H-Toeplitz operators and the commuting problem of H-Toeplitz and Hankel operators on the Bergman space. We establish necessary and sufficient conditions for the product of an H-Toeplitz operator and a Hankel operator equals another H-Toeplitz (Hankel) operator for a certain class of symbols. Moreover, we study the spectrum of the H-Toeplitz operators $ B_{z^{N}} $ and $ B_{\overline{z}^{N}} $, where $ N $ is a non-negative integer.
- H-Toeplitz operator,
- Hankel operator,
- Bergman space,
- algebraic properties,
- spectral properties
Citation: Jie Zhang. Algebraic and spectral properties of H-Toeplitz operators on the Bergman space[J]. AIMS Mathematics, 2025, 10(6): 13432-13450. doi: 10.3934/math.2025603

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Abstract

In this paper, we discuss the zero product problem of two H-Toeplitz operators and the commuting problem of H-Toeplitz and Hankel operators on the Bergman space. We establish necessary and sufficient conditions for the product of an H-Toeplitz operator and a Hankel operator equals another H-Toeplitz (Hankel) operator for a certain class of symbols. Moreover, we study the spectrum of the H-Toeplitz operators $ B_{z^{N}} $ and $ B_{\overline{z}^{N}} $, where $ N $ is a non-negative integer.

References

[1]	S. Axler, Ž. Čučković, N. V. Rao, Commutants of analytic Toeplitz operators on the Bergman space, Proc. Amer. Math. Soc., 128 (2000), 1951–1953. https://doi.org/10.1090/S0002-9939-99-05436-2 doi: 10.1090/S0002-9939-99-05436-2
[2]	S. Axler, J. B. Conway, G. McDonald, Toeplitz operators on Bergman space, Can. J. Math., 34 (1982), 466–483. https://doi.org/10.4153/CJM-1982-031-1 doi: 10.4153/CJM-1982-031-1
[3]	Q. Ding, Commuting Toeplitz operators and H-Toeplitz operators on the Bergman space, AIMS Mathematics, 9 (2023), 2530–2548. https://doi.org/10.3934/math.2024125 doi: 10.3934/math.2024125
[4]	Q. Ding, Y. Chen, Product of H-Toeplitz operator and Toeplitz operator on the Bergman space, AIMS Mathematics, 8 (2023), 20790–20801. https://doi.org/10.3934/math.20231059 doi: 10.3934/math.20231059
[5]	N. Guan, X. Zhao, Invertibility of Bergman-Toeplitz operators with harmonic polynomial symbols, Sci. China Math., 63 (2020), 965–978. https://doi.org/10.1007/s11425-018-9469-1 doi: 10.1007/s11425-018-9469-1
[6]	K. Guo, X. Zhao, D. Zheng, The spectral picture of Bergman-Toeplitz operators with harmonic polynomial symbols, Ark. Mat., 61 (2023), 343–374. https://doi.org/10.4310/ARKIV.2023.v61.n2.a5 doi: 10.4310/ARKIV.2023.v61.n2.a5
[7]	A. Gupta, S. K. Singh, H-Toeplitz operators on the Bergman spaces, Bull. Korean Math. Soc., 58 (2021), 327–347. https://doi.org/10.4134/BKMS.b200260 doi: 10.4134/BKMS.b200260
[8]	S. Kim, E. Ko, J. E. Lee, J. Lee, H-Toeplitz operators on the function spaces, Monatsh. Math., 205 (2024), 757–786. https://doi.org/10.1007/s00605-024-01985-9 doi: 10.1007/s00605-024-01985-9
[9]	S. Kim, J. Lee, Contractivity and expansivity of H-Toeplitz operators on the Bergman spaces, AIMS Mathematics, 7 (2022), 13927–13944. https://doi.org/10.3934/math.2022769 doi: 10.3934/math.2022769
[10]	J. Liang, L. Lai, Y. Zhao, Y. Chen, Commuting H-Toeplitz operators with quasihomogeneous symbols, AIMS Mathematics, 7 (2022), 7898–7908. https://doi.org/10.3934/math.2022442 doi: 10.3934/math.2022442
[11]	Y. Lu, C. Liu, Commutativity and hyponormality of Toeplitz operators on the weighted Bergman space, J. Korean Math. Soc., 46 (2009), 621–642. https://doi.org/10.4134/JKMS.2009.46.3.621 doi: 10.4134/JKMS.2009.46.3.621
[12]	H. Sadraoui, M. Guediri, Hyponormal Toeplitz operators on the Bergman space, Oper. Matrices, 11 (2017), 669–677. https://doi.org/10.7153/oam-11-44 doi: 10.7153/oam-11-44
[13]	C. Sundberg, D. Zheng, The spectrum and essential spectrum of Toeplitz operators with harmonic symbols, Indiana Univ. Math. J., 59 (2010), 385–394. https://doi.org/10.1512/iumj.2010.59.3799 doi: 10.1512/iumj.2010.59.3799
[14]	T. Hungerford, Algebra, New York: Springer, 1974. https://doi.org/10.1007/978-1-4612-6101-8
[15]	R. Yoneda, Invertibility of Toeplitz operators on the Bergman spaces with harmonic symbols, J. Math. Anal. Appl., 516 (2022), 126515. https://doi.org/10.1016/j.jmaa.2022.126515 doi: 10.1016/j.jmaa.2022.126515
[16]	X. Zhao, D. Zheng, Invertibility of Toeplitz operators via Berezin transforms, J. Operat. Theor., 75 (2016), 475–495. https://doi.org/10.7900/jot.2015jul07.2082 doi: 10.7900/jot.2015jul07.2082
[17]	X. Zhao, D. Zheng, The spectrum of Bergman Toeplitz operators with some harmonic symbols, Sci. China Math., 59 (2016), 731–740. https://doi.org/10.1007/s11425-015-5083-4 doi: 10.1007/s11425-015-5083-4
[18]	K. Zhu, Hankel operators on the Bergman space of bounded symmetric domains, Trans. Amer. Math. Soc., 324 (1991), 707–730. https://doi.org/10.1090/S0002-9947-1991-1093426-6 doi: 10.1090/S0002-9947-1991-1093426-6
[19]	K. Zhu, Operator theory in function spaces, 2 Eds., Providence: American Mathematical Society, 2007. https://doi.org/10.1090/surv/138

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