Abstract
In this paper, we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space. First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal. Then we obtain a necessary and sufficient condition for the dual Toeplitz operator Sφ with the symbol \(\varphi(z)=az^{{n_{1}-m_{1}}}\!\!\!\!\!\!\!\!\!z+bz^{{n_{2}-m_{2}}}\!\!\!\!\!\!\!\!\!z\) (n1, n2, m1, \(m_{2}\in\mathbb{N}\) and a, b ∈ ℂ) to be hyponormal. Finally, we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank.
This is a preview of subscription content, log in via an institution to check access.
References
Arveson, W.: A Short Course on Spectral Theory, Graduate Texts in Mathematics, Vol. 209, Springer-Verlag, New York, 2000
Brown, A., Halmos, P. R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math., 213, 89–102 (1964)
Chen, Y., Koo, H., Lee, Y. J.: Ranks of commutators of Toeplitz operators on the harmonic Bergman space. Integral Equations Operator Theory, 75, 31–38 (2013)
Chen, Y., Koo, H., Lee, Y. J.: Ranks of complex skew symmetric operators and applications to Toeplitz operators. J. Math. Anal. Appl., 425, 734–747 (2015)
Chen, Y., Yu, T., Zhao, Y.: Dual Toeplitz operators on orthogonal complement of the harmonic Dirichlet space. Acta Math. Sin., Engl. Ser., 33, 383–402 (2017)
Choe, B. R., Lee, Y. J.: Commuting Toeplitz operators on the harmonic Bergman space. Michigan Math. J., 46, 163–174 (1999)
Conway, J. B.: A Course in Functional Analysis, Second Edition, Graduate Texts in Mathematics, Vol. 96, Springer-Verlag, New York, 1990.
Cowen, C. C.: Hyponormality of Toeplitz operators. Proc. Amer. Math. Soc., 103, 809–812 (1998)
Čučković, Ž., Curto, R. E.: A new necessary condition for the hyponormality of Toeplitz operators on the Bergman space. J. Operator Theory, 79, 287–300 (2018)
Curto, R. E., Hwang, I. S., Lee, W. Y.: Hyponormality and subnormality of block Toeplitz operators. Adv. Math., 230, 2094–2151 (2012)
Ding, X., Wu, S., Zhao, X.: Invertibility and spectral properties of dual Toeplitz operators. J. Math. Anal. Appl., 484, 1–18 (2020)
Douglas, R.: Banach Algebra Techniques in Operator Theory, Second Edition, Graduate Texts in Mathematics, Vol. 179, Springer-Verlag, New York, 1998
Farenick, D. R., Lee, W. Y.: Hyponormality and spectra of Toeplitz operators. Trans. Amer. Math. Soc., 348, 4153–4174 (1996)
Hwang, I. S., Lee, J.: Hyponormal Toeplitz operators on the Bergman space II. Bull. Korean Math. Soc., 44, 517–522 (2007)
Hwang, I. S.: Hyponormal Toeplitz operators on the Bergman spaces. J. Korean Math. Soc., 45, 1027–1041 (2008)
Hwang, I. S., Kim, I. H., Lee, W. Y.: Hyponormality of Toeplitz operators with polynomial symbols. Math. Ann., 313, 247–261 (1999)
Le, T., Simanek, B.: Hyponormality of Toeplitz operators on weighted Bergman space, Integral Transforms Spec. Funct., 32, 560–567 (2021)
Lu, Y.: Commuting dual Toeplitz operators with pluriharmonic symbols. J. Math. Anal. Appl., 302, 149–156 (2005)
Lu, Y., Shang, S.: Commuting dual Toeplitz operators on the polydisk. Acta Math. Sin., Engl. Ser., 23, 857–868 (2007)
Lu, Y., Shi, Y.: Hyponormal Toeplitz operators on the weighted Bergman space. Integral Equations Operator Theory, 65, 115–129 (2009)
Nakaz, T., Takahashi, K.: Hyponormal Toeplitz operators and extremal problems of Hardy spaces. Trans. Amer. Math. Soc., 338, 753–766 (1993)
Peng, Y., Zhao, X.: Dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space. Acta Math. Sin., Engl. Ser., 37, 1143–1155 (2021)
Stroethoff, K., Zheng, D.: Products of Hankel and Toeplitz operators on the Bergman space. J. Funct. Anal., 169, 289–313 (1999)
Stroethoff, K., Zheng, D.: Algebraic and spectral properties of dual Toeplitz operators. Trans. Amer. Math. Soc., 354, 2495–2520 (2002)
Yang, J., Hu, Y., Lu, Y., Yu, T.: Commuting dual Toeplitz operators on the harmonic Dirichlet space. Acta Math. Sin., Engl. Ser., 32, 1099–1105 (2016)
Yang, Y., Lu, Y.: Commuting dual Toeplitz operators on the harmonic Bergman space. Sci. China Math., 58, 1461–1472 (2015)
Yu, D.: Hyponormal Toeplitz operators on \(H^{2}(\mathbb{T})\) with polynomial symbols. Nagoya Math. J., 144, 179–182 (1996)
Yu, T., Wu, S.: Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space. Acta Math. Sin., Engl. Ser., 252, 245–252 (2009)
Zhu, K.: Hyponormal Toeplitz operators with polynomial symbols. Integral Equations Operator Theory, 21, 376–381 (1995)
Zhu, K.: Operator Theory in Function Spaces, Second Edition, Mathematical Surveys and Monographs, Vol. 138, American Mathematical Society, 2007
Acknowledgements
We thank the reviewers for providing constructive comments and helping to improve the content of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by NSFC (Grant No. 11701052); the second author was partially supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 2020CDJQY-A039 and 2020CDJ-LHSS-003)
Rights and permissions
About this article
Cite this article
Wang, C.C., Zhao, X.F. Hyponormal Dual Toeplitz Operators on the Orthogonal Complement of the Harmonic Bergman Space. Acta. Math. Sin.-English Ser. 39, 846–862 (2023). https://doi.org/10.1007/s10114-023-1382-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-023-1382-9