Skip to main content Accesibility Help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article

Commuting and Semi-commuting Monomial-type Toeplitz Operators on Some Weakly Pseudoconvex Domains

  • Cao Jiang (a1), Xing-Tang Dong (a1) and Ze-Hua Zhou (a1)
Abstract

In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.

Copyright
© Canadian Mathematical Society 2018 
Footnotes
Hide All

Z.-H. Zhou is corresponding author. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11201331; 11371276; 11771323).

Footnotes
References
Hide All
[1] Abate, M., Raissy, J., and Saracco, A., Toeplitz operators and Carleson measures in strongly pseudoconvex domains . J. Funct. Anal. 263(2012), 3449–3491. https://doi.org/10.1016/j.jfa.2012.08.027.
[2] Ahern, P. and Čučković, Ž., A theorem of Brown-Halmos type for Bergman space Toeplitz operators . J. Funct. Anal. 187(2001), 200–210. https://doi.org/10.1006/jfan.2001.3811.
[3] Axler, S. and Čučković, Ž., Commuting Toeplitz operators with harmonic symbols . Integr. Equat. Oper. Theory. 14(1991), 1–12. https://doi.org/10.1007/BF01194925.
[4] Crocker, D. and Raeburn, I., Toeplitz operators on certain weakly pseudoconvex domains . J. Austral. Math. Soc. Ser. A. 31(1981), 1–14. https://doi.org/10.1017/S1446788700018437.
[5] Čučković, Ž. and Rao, N. V., Mellin transform, monomial symbols, and commuting Toeplitz operators . J. Funct. Anal. 154(1998), 195–214. https://doi.org/10.1006/jfan.1997.3204.
[6] Dong, X.-T. and Zhou, Z.-H., Algebraic properties of Toeplitz operators with separately quasihomogeneous symbols on the Bergman space of the unit ball . J. Operator Theory 66(2011), 193–207.
[7] Dong, X.-T. and Zhou, Z.-H., Ranks of commutators and generalized semi-commutators of quasi-homogeneous Toeplitz operators . Monatsh. Math. 183(2017), 103–141. https://doi.org/10.1007/s00605-017-1033-2.
[8] Dong, X.-T. and Zhu, K., Commutators and semi-commutators of Toeplitz operators on the unit ball . Integral Equations Operator Theory 86(2016), 271–300. https://doi.org/10.1007/s00020-016-2326-x.
[9] Guo, K., Sun, S., and Zheng, D., Finite rank commutators and semicommutators of Toeplitz operators with harmonic symbols . Illinois J. Math. 51(2007), 583–596.
[10] Jewell, N. P. and Krantz, S. G., Toeplitz operators and related function algebras on certain pseudoconvex domains . Trans. Amer. Math. Soc. 252(1979), 297–312. https://doi.org/10.1090/S0002-9947-1979-0534123-7.
[11] Le, T., The commutants of certain Toeplitz operators on weighted Bergman spaces . J. Math. Anal. Appl. 348(2008), 1–11. https://doi.org/10.1016/j.jmaa.2008.07.005.
[12] Le, T., Commutants of separately radial Toeplitz operators in several variables . J. Math. Anal. Appl. 453(2017), 48–63. https://doi.org/10.1016/j.jmaa.2017.03.088.
[13] Louhichi, I., Strouse, E., and Zakariasy, L., Products of Toeplitz operators on the Bergman space . Integral Equations Operator Theory 54(2006), 525–539. https://doi.org/10.1007/s00020-005-1369-1.
[14] Louhichi, I. and Zakariasy, L., On Toeplitz operators with quasihomogeneous symbols . Arch. Math. 85(2005), 248–257. https://doi.org/10.1007/s00013-005-1198-0.
[15] Quiroga-Barranco, R. and Sanchez-Nungaray, A., Toeplitz operators with quasi-Homogeneous quasi-radial symbols on some weakly pseudoconvex domains . Complex Anal. Oper. Theory 9(2015), 1111–1134. https://doi.org/10.1007/s11785-014-0407-x.
[16] Vasilevski, N., Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators . Integral Equations Operator Theory 66(2010), 141–152. https://doi.org/10.1007/s00020-009-1732-8.
[17] Webster, S. M., Biholomorphic mappings and the Bergmann kernel off the diagonal . Invent. Math. 51(1979), 155–169. https://doi.org/10.1007/BF01390226.
[18] Zheng, D., Commuting Toeplitz Operators with Pluriharmonic Symbols . Trans. Amer. Math. Soc. 350(1998), 1595–1618. https://doi.org/10.1090/S0002-9947-98-02051-0.
[19] Zhou, Z. H. and Dong, X. T., Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball . Integral Equations Operator Theory 64(2009), 137–154. https://doi.org/10.1007/s00020-009-1677-y.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

Cancel
Confirm
×