×

Algebraic properties of Toeplitz operators on the polydisk. (English) Zbl 1231.32003

Summary: We discuss some algebraic properties of Toeplitz operators on the Bergman space of the polydisk \(\mathbb D^n\). First, we introduce Toeplitz operators with quasihomogeneous symbols and property (P). Secondly, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. Thirdly, we discuss finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols. Finally, we solve the finite rank product problem for Toeplitz operators on the polydisk.

MSC:

32A36 Bergman spaces of functions in several complex variables
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Z. H. Zhou and X. T. Dong, “Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball,” Integral Equations and Operator Theory, vol. 64, no. 1, pp. 137-154, 2009. · Zbl 1195.47020 · doi:10.1007/s00020-009-1677-y
[2] A. Brown and P. R. Halmos, “Algebraic properties of Toeplitz operators,” Journal für die Reine und Angewandte Mathematik, vol. 213, pp. 89-102, 1963. · Zbl 0116.32501
[3] S. Axler, \vZ. \vCu, and N. V. Rao, “Commutants of analytic Toeplitz operators on the Bergman space,” Proceedings of the American Mathematical Society, vol. 128, no. 7, pp. 1951-1953, 2000. · Zbl 0947.47023 · doi:10.1090/S0002-9939-99-05436-2
[4] \vZ. \vCu, “Commutants of Toeplitz operators on the Bergman space,” Pacific Journal of Mathematics, vol. 162, no. 2, pp. 277-285, 1994. · Zbl 0802.47018 · doi:10.2140/pjm.1994.162.277
[5] \vZ. \vCu and I. Louhichi, “Finite rank commutators and semicommutators of quasihomogeneous Toeplitz operators,” Complex Analysis and Operator Theory, vol. 2, no. 3, pp. 429-439, 2008. · Zbl 1183.47020 · doi:10.1007/s11785-007-0044-8
[6] \vZ. \vCu and N. V. Rao, “Mellin transform, monomial symbols, and commuting Toeplitz operators,” Journal of Functional Analysis, vol. 154, no. 1, pp. 195-214, 1998. · Zbl 0936.47015 · doi:10.1006/jfan.1997.3204
[7] I. Louhichi and L. Zakariasy, “On Toeplitz operators with quasihomogeneous symbols,” Archiv der Mathematik, vol. 85, no. 3, pp. 248-257, 2005. · Zbl 1088.47019 · doi:10.1007/s00013-005-1198-0
[8] B. R. Choe and Y. J. Lee, “Pluriharmonic symbols of commuting Toeplitz operators,” Illinois Journal of Mathematics, vol. 37, no. 3, pp. 424-436, 1993. · Zbl 0816.47024
[9] T. Le, “The commutants of certain Toeplitz operators on weighted Bergman spaces,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 1-11, 2008. · Zbl 1168.47025 · doi:10.1016/j.jmaa.2008.07.005
[10] Y. Lu and B. Zhang, “Finite rank commutator of Toeplitz operators with quasihomogeneous symbols on the unit ball,” Acta Mathematica SinicIn press.
[11] D. Zheng, “Commuting Toeplitz operators with pluriharmonic symbols,” Transactions of the American Mathematical Society, vol. 350, no. 4, pp. 1595-1618, 1998. · Zbl 0893.47015 · doi:10.1090/S0002-9947-98-02051-0
[12] D. Xuanhao and T. Shengqiang, “The pluriharmonic Toeplitz operators on the polydisk,” Journal of Mathematical Analysis and Applications, vol. 254, no. 1, pp. 233-246, 2001. · Zbl 0987.47015 · doi:10.1006/jmaa.2000.7265
[13] Y. J. Lee, “Commuting Toeplitz operators on the Hardy space of the bidisk,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 738-749, 2008. · Zbl 1138.47024 · doi:10.1016/j.jmaa.2007.11.002
[14] Y. J. Lee, “Commuting Toeplitz operators on the Hardy space of the polydisk,” Proceedings of the American Mathematical Society, vol. 138, no. 1, pp. 189-197, 2010. · Zbl 1189.47028 · doi:10.1090/S0002-9939-09-10073-4
[15] D. H. Luecking, “Finite rank Toeplitz operators on the Bergman space,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1717-1723, 2008. · Zbl 1152.47021 · doi:10.1090/S0002-9939-07-09119-8
[16] B. R. Choe, “On higher dimensional Luecking’s theorem,” Journal of the Mathematical Society of Japan, vol. 61, no. 1, pp. 213-224, 2009. · Zbl 1162.47027 · doi:10.2969/jmsj/06110213
[17] T. Le, “Finite-rank products of Toeplitz operators in several complex variables,” Integral Equations and Operator Theory, vol. 63, no. 4, pp. 547-555, 2009. · Zbl 1228.47031 · doi:10.1007/s00020-009-1661-6
[18] T. Le, “A refined Luecking’s theorem and finite-rank products of Toeplitz operators,” Complex Analysis and Operator Theory, vol. 4, no. 2, pp. 391-399, 2010. · Zbl 1226.47024 · doi:10.1007/s11785-009-0008-2
[19] T. Le, “Compact Hankel operators on generalized Bergman spaces of the polydisc,” Integral Equations and Operator Theory, vol. 67, no. 3, pp. 425-438, 2010. · Zbl 1237.47031 · doi:10.1007/s00020-010-1788-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.