Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
complex symmetry; Toeplitz operator; weighted Bergman space
complex symmetry; Toeplitz operator; weighted Bergman space
Summary:
We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^p \bar {z}^q}$ on the weighted Bergman space $A^2(\Omega )$, where $\Omega $ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^p \bar {z}^q}$ to be complex symmetric. When $p,q \in \mathbb {N}^2$, we prove that $T_{z^p \bar {z}^q}$ is complex symmetric on $A^2(\Omega )$ if and only if $p_1 = q_2$ and $p_2 = q_1$. Moreover, we completely characterize when monomial Toeplitz operators $T_{z^p \bar {z}^q}$ on $A^2(\mathbb {D}_{n})$ are $J_U$-symmetric with the $ n \times n$ symmetric unitary matrix $U$.
References:
[1] Bu, Q., Chen, Y., Zhu, S.: Complex symmetric Toeplitz operators. Integral Equations Oper. Theory 93 (2021), Article ID 15, 19 pages. DOI 10.1007/s00020-021-02629-5 | MR 4233225 | Zbl 07345264
[2] Dong, X.-T., Zhu, K.: Commutators and semi-commutators of Toeplitz operators on the unit ball. Integral Equations Oper. Theory 86 (2016), 271-300. DOI 10.1007/s00020-016-2326-x | MR 3568017 | Zbl 1448.47038
[3] Garcia, S. R., Poore, D. E.: On the norm closure problem for complex symmetric operators. Proc. Am. Math. Soc. 141 (2013), 549. DOI 10.1090/S0002-9939-2012-11347-4 | MR 2996959 | Zbl 1264.47003
[4] Garcia, S. R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358 (2006), 1285-1315. DOI 10.1090/S0002-9947-05-03742-6 | MR 2187654 | Zbl 1087.30031
[5] Garcia, S. R., Putinar, M.: Complex symmetric operators and applications. II. Trans. Am. Math. Soc. 359 (2007), 3913-3931. DOI 10.1090/S0002-9947-07-04213-4 | MR 2302518 | Zbl 1123.47030
[6] Garcia, S. R., Wogen, W. R.: Complex symmetric partial isometries. J. Funct. Anal. 257 (2009), 1251-1260. DOI 10.1016/j.jfa.2009.04.005 | MR 2535469 | Zbl 1166.47023
[7] Garcia, S. R., Wogen, W. R.: Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 362 (2010), 6065-6077. DOI 10.1090/S0002-9947-2010-05068-8 | MR 2661508 | Zbl 1208.47036
[8] Guo, K., Ji, Y., Zhu, S.: A $C^*$-algebra approach to complex symmetric operators. Trans. Am. Math. Soc. 367 (2015), 6903-6942. DOI 10.1090/S0002-9947-2015-06215-1 | MR 3378818 | Zbl 1393.47018
[9] Guo, K., Zhu, S.: A canonical decomposition of complex symmetric operators. J. Oper. Theory 72 (2014), 529-547. DOI 10.7900/jot.2013aug15.2007 | MR 3272045 | Zbl 1389.47042
[10] Jiang, C., Dong, X., Zhou, Z.: Complex symmetric Toeplitz operators on the unit polydisk and the unit ball. Acta Math. Sci., Ser. B, Engl. Ed. 40 (2020), 35-44. DOI 10.1007/s10473-020-0103-2 | MR 4070746
[11] Jiang, C., Zhou, Z.-H., Dong, X.-T.: Commutator and semicommutator of two monomial-type Toeplitz operators on the unit polydisk. Complex Anal. Oper. Theory 13 (2019), 2095-2121. DOI 10.1007/s11785-017-0730-0 | MR 3979700 | Zbl 07081947
[12] Ko, E., Lee, J. E.: On complex symmetric Toeplitz operators. J. Math. Anal. Appl. 434 (2016), 20-34. DOI 10.1016/j.jmaa.2015.09.004 | MR 3404546 | Zbl 1347.47019
[13] Li, A., Liu, Y., Chen, Y.: Complex symmetric Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl. 487 (2020), Article ID 123998, 12 pages. DOI 10.1016/j.jmaa.2020.123998 | MR 4074195 | Zbl 1435.47041
[14] Li, R., Yang, Y., Lu, Y.: A class of complex symmetric Toeplitz operators on Hardy and Bergman spaces. J. Math. Anal. Appl. 489 (2020), Article ID 124173, 11 pages. DOI 10.1016/j.jmaa.2020.124173 | MR 4093056 | Zbl 07205245
[15] Noor, S. W.: Complex symmetry of Toeplitz operators with continuous symbols. Arch. Math. 109 (2017), 455-460. DOI 10.1007/s00013-017-1101-9 | MR 3710765 | Zbl 06798496
[16] Zhu, S., Li, C. G.: Complex symmetric weighted shifts. Trans. Am. Math. Soc. 365 (2013), 511-530. DOI 10.1090/S0002-9947-2012-05642-X | MR 2984066 | Zbl 1282.47045
[17] Zhu, S., Li, C. G., Ji, Y. Q.: The class of complex symmetric operators is not norm closed. Proc. Am. Math. Soc. 140 (2012), 1705-1708. DOI 10.1090/S0002-9939-2011-11345-5 | MR 2869154 | Zbl 1251.47004
Search
Partner of
