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Keywords:
Bergman space; Toeplitz operator; Hankel operator; Berezin transform
Summary:
We study algebraic properties of two Toeplitz operators on the weighted Bergman space on the unit disk with harmonic symbols. In particular the product property and commutative property are discussed. Further we apply our results to solve a compactness problem of the product of two Hankel operators on the weighted Bergman space on the unit bidisk.
References:
[1] Ahern, P.: On the range of the Berezin transform. J. Funct. Anal. 215 (2004), 206-216. DOI 10.1016/j.jfa.2003.08.007 | MR 2085115 | Zbl 1088.47014
[2] Ahern, P., Čučković, Ž.: A theorem of Brown-Halmos type for Bergman space Toeplitz operators. J. Funct. Anal. 187 (2001), 200-210. DOI 10.1006/jfan.2001.3811 | MR 1867348 | Zbl 0996.47037
[3] Ahern, P., Flores, M., Rudin, W.: An invariant volume-mean-value property. J. Funct. Anal. 111 (1993), 380-397. DOI 10.1006/jfan.1993.1018 | MR 1203459 | Zbl 0771.32006
[4] Axler, S., Čučković, Ž.: Commuting Toeplitz operators with harmonic symbols. Integral Equations Oper. Theory 14 (1991), 1-12. DOI 10.1007/BF01194925 | MR 1079815 | Zbl 0733.47027
[5] Axler, S., Čučković, Ž., Rao, N. V.: Commutants of analytic Toeplitz operators on the Bergman space. Proc. Am. Math. Soc. 128 (2000), 1951-1953. DOI 10.1090/S0002-9939-99-05436-2 | MR 1694299 | Zbl 0947.47023
[6] Brown, A., Halmos, P. R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213 (1963/64), 89-102. DOI 10.1515/crll.1964.213.89 | MR 0160136 | Zbl 0116.32501
[7] Choe, B. R., Lee, Y. J., Nam, K., Zheng, D.: Products of Bergman space Toeplitz operators on the polydisk. Math. Ann. 337 (2007), 295-316. DOI 10.1007/s00208-006-0034-6 | MR 2262785 | Zbl 1122.47022
[8] Čučković, Ž., Li, B.: Berezin transform, Mellin transform and Toeplitz operators. Complex Anal. Oper. Theory 6 (2012), 189-218. DOI 10.1007/s11785-010-0051-z | MR 2886615 | Zbl 1303.47036
[9] Čučković, Ž., Rao, N. V.: Mellin transform, monomial symbols, and commuting Toeplitz operators. J. Funct. Anal. 154 (1998), 195-214. DOI 10.1006/jfan.1997.3204 | MR 1616532 | Zbl 0936.47015
[10] Čučković, Ž., Şahutoğlu, S.: Compactness of products of Hankel operators on the polydisk and some product domains in $C^2$. J. Math. Anal. Appl. 371 (2010), 341-346. DOI 10.1016/j.jmaa.2010.05.027 | MR 2661011 | Zbl 1200.32001
[11] Rao, N. V.: Range of Berezin transform. Available at https://arxiv.org/abs/1003.3939 (2010), 10 pages.
[12] Zhu, K.: Operator Theory in Function Spaces. Mathematical Surveys and Monographs 138. American Mathematical Society, Providence (2007). DOI 10.1090/surv/138 | MR 2311536 | Zbl 1123.47001
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