Complex symmetry of Toeplitz operators on the weighted Bergman spaces

Xiao-He Hu

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 3, page 855-873
  • ISSN: 0011-4642

Abstract

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We give a concrete description of complex symmetric monomial Toeplitz operators T z p z ¯ q on the weighted Bergman space A 2 ( Ω ) , where Ω denotes the unit ball or the unit polydisk. We provide a necessary condition for T z p z ¯ q to be complex symmetric. When p , q ∈ ℕ 2 , we prove that T z p z ¯ q is complex symmetric on A 2 ( Ω ) if and only if p 1 = q 2 and p 2 = q 1 . Moreover, we completely characterize when monomial Toeplitz operators T z p z ¯ q on A 2 ( 𝔻 n ) are J U -symmetric with the n × n symmetric unitary matrix U .

How to cite

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Hu, Xiao-He. "Complex symmetry of Toeplitz operators on the weighted Bergman spaces." Czechoslovak Mathematical Journal 72.3 (2022): 855-873. <http://eudml.org/doc/298381>.

@article{Hu2022,
abstract = {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 $U$.},
author = {Hu, Xiao-He},
journal = {Czechoslovak Mathematical Journal},
keywords = {complex symmetry; Toeplitz operator; weighted Bergman space},
language = {eng},
number = {3},
pages = {855-873},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Complex symmetry of Toeplitz operators on the weighted Bergman spaces},
url = {http://eudml.org/doc/298381},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Hu, Xiao-He
TI - Complex symmetry of Toeplitz operators on the weighted Bergman spaces
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 3
SP - 855
EP - 873
AB - 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 $U$.
LA - eng
KW - complex symmetry; Toeplitz operator; weighted Bergman space
UR - http://eudml.org/doc/298381
ER -

References

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  1. Bu, Q., Chen, Y., Zhu, S., 10.1007/s00020-021-02629-5, Integral Equations Oper. Theory 93 (2021), Article ID 15, 19 pages. (2021) Zbl07345264MR4233225DOI10.1007/s00020-021-02629-5
  2. Dong, X.-T., Zhu, K., 10.1007/s00020-016-2326-x, Integral Equations Oper. Theory 86 (2016), 271-300. (2016) Zbl1448.47038MR3568017DOI10.1007/s00020-016-2326-x
  3. Garcia, S. R., Poore, D. E., 10.1090/S0002-9939-2012-11347-4, Proc. Am. Math. Soc. 141 (2013), 549. (2013) Zbl1264.47003MR2996959DOI10.1090/S0002-9939-2012-11347-4
  4. Garcia, S. R., Putinar, M., 10.1090/S0002-9947-05-03742-6, Trans. Am. Math. Soc. 358 (2006), 1285-1315. (2006) Zbl1087.30031MR2187654DOI10.1090/S0002-9947-05-03742-6
  5. Garcia, S. R., Putinar, M., 10.1090/S0002-9947-07-04213-4, Trans. Am. Math. Soc. 359 (2007), 3913-3931. (2007) Zbl1123.47030MR2302518DOI10.1090/S0002-9947-07-04213-4
  6. Garcia, S. R., Wogen, W. R., 10.1016/j.jfa.2009.04.005, J. Funct. Anal. 257 (2009), 1251-1260. (2009) Zbl1166.47023MR2535469DOI10.1016/j.jfa.2009.04.005
  7. Garcia, S. R., Wogen, W. R., 10.1090/S0002-9947-2010-05068-8, Trans. Am. Math. Soc. 362 (2010), 6065-6077. (2010) Zbl1208.47036MR2661508DOI10.1090/S0002-9947-2010-05068-8
  8. Guo, K., Ji, Y., Zhu, S., 10.1090/S0002-9947-2015-06215-1, Trans. Am. Math. Soc. 367 (2015), 6903-6942. (2015) Zbl1393.47018MR3378818DOI10.1090/S0002-9947-2015-06215-1
  9. Guo, K., Zhu, S., 10.7900/jot.2013aug15.2007, J. Oper. Theory 72 (2014), 529-547. (2014) Zbl1389.47042MR3272045DOI10.7900/jot.2013aug15.2007
  10. Jiang, C., Dong, X., Zhou, Z., 10.1007/s10473-020-0103-2, Acta Math. Sci., Ser. B, Engl. Ed. 40 (2020), 35-44. (2020) MR4070746DOI10.1007/s10473-020-0103-2
  11. Jiang, C., Zhou, Z.-H., Dong, X.-T., 10.1007/s11785-017-0730-0, Complex Anal. Oper. Theory 13 (2019), 2095-2121. (2019) Zbl07081947MR3979700DOI10.1007/s11785-017-0730-0
  12. Ko, E., Lee, J. E., 10.1016/j.jmaa.2015.09.004, J. Math. Anal. Appl. 434 (2016), 20-34. (2016) Zbl1347.47019MR3404546DOI10.1016/j.jmaa.2015.09.004
  13. Li, A., Liu, Y., Chen, Y., 10.1016/j.jmaa.2020.123998, J. Math. Anal. Appl. 487 (2020), Article ID 123998, 12 pages. (2020) Zbl1435.47041MR4074195DOI10.1016/j.jmaa.2020.123998
  14. Li, R., Yang, Y., Lu, Y., 10.1016/j.jmaa.2020.124173, J. Math. Anal. Appl. 489 (2020), Article ID 124173, 11 pages. (2020) Zbl07205245MR4093056DOI10.1016/j.jmaa.2020.124173
  15. Noor, S. W., 10.1007/s00013-017-1101-9, Arch. Math. 109 (2017), 455-460. (2017) Zbl06798496MR3710765DOI10.1007/s00013-017-1101-9
  16. Zhu, S., Li, C. G., 10.1090/S0002-9947-2012-05642-X, Trans. Am. Math. Soc. 365 (2013), 511-530. (2013) Zbl1282.47045MR2984066DOI10.1090/S0002-9947-2012-05642-X
  17. Zhu, S., Li, C. G., Ji, Y. Q., 10.1090/S0002-9939-2011-11345-5, Proc. Am. Math. Soc. 140 (2012), 1705-1708. (2012) Zbl1251.47004MR2869154DOI10.1090/S0002-9939-2011-11345-5

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