This paper aimed to give some partial answers to the zero-product problem and commutativity problem concerning dual Toeplitz operators with nonharmonic symbols on the orthogonal complement of the harmonic Bergman space. Using the symbol map, we described the necessary condition for $ S_{\varphi_1}S_{\varphi_2}\cdots S_{\varphi_N} = 0 $ with radial symbols. Furthermore, we established the sufficient and necessary conditions for $ S_{\varphi}S_{ \psi} = S_{\psi}S_{\varphi} $ with $ \varphi(z) = az^{p_1}\overline{z}^{q_1}+bz^{p_2}\overline{z}^{q_2} $ and $ \psi(z) = z^{s}\overline{z}^{t} $.
Citation: Lijun Liu. Dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space[J]. AIMS Mathematics, 2024, 9(9): 25413-25437. doi: 10.3934/math.20241241
This paper aimed to give some partial answers to the zero-product problem and commutativity problem concerning dual Toeplitz operators with nonharmonic symbols on the orthogonal complement of the harmonic Bergman space. Using the symbol map, we described the necessary condition for $ S_{\varphi_1}S_{\varphi_2}\cdots S_{\varphi_N} = 0 $ with radial symbols. Furthermore, we established the sufficient and necessary conditions for $ S_{\varphi}S_{ \psi} = S_{\psi}S_{\varphi} $ with $ \varphi(z) = az^{p_1}\overline{z}^{q_1}+bz^{p_2}\overline{z}^{q_2} $ and $ \psi(z) = z^{s}\overline{z}^{t} $.
| [1] | K. Zhu, Operator theory in function spaces, 2 Eds., American Mathematical Society, 2007. https://doi.org/10.1090/surv/138 |
| [2] |
B. Choe, Y. Lee, Commuting Toeplitz operators on the harmonic Bergman space, Michigan Math. J., 46 (1999), 163–174. https://doi.org/10.1307/MMJ/1030132367 doi: 10.1307/MMJ/1030132367
|
| [3] |
K. Sttroethoff, D. Zheng, Algebraic and spectral properties of dual Toeplitz operators, Trans. Amer. Math. Soc., 354 (2002), 2495–2520. https://doi.org/10.1090/S0002-9947-02-02954-9 doi: 10.1090/S0002-9947-02-02954-9
|
| [4] |
K. Stroethoff, D. Zheng, Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal., 169 (1999), 289–313. https://doi.org/10.1006/jfan.1999.3489 doi: 10.1006/jfan.1999.3489
|
| [5] |
Y. Hu, Y. Lu, Y. Yang, Properties of dual Toeplitz operators on the orthogonal complement of the pluriharmonic Bergman space of the unit ball, Fac. Sci. Math., 36 (2022), 4265–4276. https://doi.org/10.2298/fil2212265h doi: 10.2298/fil2212265h
|
| [6] |
Y. Lu, Commuting dual Toeplitz operators with pluriharmonic symbols, J. Math. Anal. Appl., 302 (2005), 149–156. https://doi.org/10.1016/j.jmaa.2004.07.046 doi: 10.1016/j.jmaa.2004.07.046
|
| [7] |
Y. Lu, J. Yang, Commuting dual Toeplitz operators on weighted Bergman spaces of the unit ball, Acta Math. Sin. Engl. Ser., 27 (2011), 1725–1742. https://doi.org/10.1007/s10114-011-8577-1 doi: 10.1007/s10114-011-8577-1
|
| [8] |
Y. Lu, S. Shang, Commuting dual Toeplitz operators on the polydisk, Acta Math. Sin. Engl. Ser., 23 (2007), 857–868. https://doi.org/10.1007/s10114-005-0877-x doi: 10.1007/s10114-005-0877-x
|
| [9] |
T. Yu, S. Wu, Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space, Acta Math. Sin., Engl. Ser., 25 (2009), 245–252. https://doi.org/10.1007/s10114-008-7109-0 doi: 10.1007/s10114-008-7109-0
|
| [10] |
T. Yu, Operators on the orthogonal complement of the Dirichlet space, J. Math. Anal. Appl., 357 (2009), 300–306. https://doi.org/10.1016/j.jmaa.2009.04.019 doi: 10.1016/j.jmaa.2009.04.019
|
| [11] |
T. Yu, Operators on the orthogonal complement of the Dirichlet space (II), Sci. China Math., 54 (2011), 2005–2012. https://doi.org/10.1007/s11425-011-4259-9 doi: 10.1007/s11425-011-4259-9
|
| [12] |
Y. Peng, X. Zhao, Dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space, Acta Math. Sin. Engl. Ser., 37 (2021), 1143–1155. https://doi.org/10.1007/s10114-021-0315-8 doi: 10.1007/s10114-021-0315-8
|
| [13] |
J. Yang, Y. Lu, Commuting dual Toeplitz operators on the harminic Bergman space, Sci. China Math., 58 (2015), 1461–1472. https://doi.org/10.1007/s11425-014-4940-x doi: 10.1007/s11425-014-4940-x
|
| [14] |
Y. Chen, T. Yu, Y. Zhao, Dual Toeplitz operators on orthogonal complement of the harmonic Dirichlet space, Acta Math. Sin. Engl. Ser., 33 (2017), 383–402. https://doi.org/10.1007/s10114-016-5779-6 doi: 10.1007/s10114-016-5779-6
|
| [15] |
C. Wang, X. Zhao, Hyponormal dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space, Acta. Math. Sin. Engl. Ser., 39 (2023), 846–862. https://doi.org/10.1007/s10114-023-1382-9 doi: 10.1007/s10114-023-1382-9
|
Lijun Liu. Dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space[J]. AIMS Mathematics, 2024, 9(9): 25413-25437. doi: 10.3934/math.20241241
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