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FINITE LINEAR COMBINATIONS OF MONOMIAL TOEPLITZ PRODUCTS ON THE BERGMAN SPACE

Part of: Special classes of linear operators Spaces and algebras of analytic functions

Published online by Cambridge University Press:  10 March 2025

KAIXUAN CAO ,
XINGTANG DONG Open the ORCID record for XINGTANG DONG [Opens in a new window]  and
RUIMENG LIU
KAIXUAN CAO
Affiliation:
School of Mathematics, Tianjin University, Tianjin 300354, PR China e-mail: [email protected]
XINGTANG DONG*
Affiliation:
School of Mathematics, Tianjin University, Tianjin 300354, PR China e-mail: [email protected]
RUIMENG LIU
Affiliation:
School of Mathematics, Tianjin University, Tianjin 300354, PR China e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

In sharp contrast to the Hardy space case, the algebraic properties of Toeplitz operators on the Bergman space are quite different and abnormally complicated. In this paper, we study the finite-rank problem for a class of operators consisting of all finite linear combinations of Toeplitz products with monomial symbols on the Bergman space of the unit disk. It turns out that such a problem is equivalent to the problem of when the corresponding finite linear combination of rational functions is zero. As an application, we consider the finite-rank problem for the commutator and semi-commutator of Toeplitz operators whose symbols are finite linear combinations of monomials. In particular, we construct many motivating examples in the theory of algebraic properties of Toeplitz operators.

Keywords

monomial Toeplitz operatorBergman spacecommutatorsemi-commutatorfinite rank

MSC classification

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Secondary: 47B47: Commutators, derivations, elementary operators, etc. 30H20: Bergman spaces, Fock spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Ji Li

The second author was supported in part by the National Natural Science Foundation of China (Grant Nos. 12271396, 12171353).

References

Ahern, P. and Čučković, Z., ‘A theorem of Brown–Halmos type for Bergman space Toeplitz operators’, J. Funct. Anal. 187 (2001), 200–210.CrossRefGoogle Scholar
Axler, S., Chang, S. and Sarason, D., ‘Product of Toeplitz operators’, Integral Equations Operator Theory 1 (1978), 285–309.CrossRefGoogle Scholar
Axler, S. and Čučković, Z., ‘Commuting Toeplitz operators with harmonic symbols’, Integral Equations Operator Theory 14 (1991), 1–12.CrossRefGoogle Scholar
Brown, A. and Halmos, P., ‘Algebraic properties of Toeplitz operators’, J. reine angew. Math. 213 (1964), 89–102.Google Scholar
Čučković, Ž., ‘Commutants of Toeplitz operators on the Bergman space’, Pacific J. Math. 162 (1994), 277–285.CrossRefGoogle Scholar
Čučković, Ž., ‘Finite rank perturbations of Toeplitz operators’, Integral Equations Operator Theory 59 (2007), 345–353.CrossRefGoogle Scholar
Čučković, Ž. and Louhichi, I., ‘Finite rank commutators and semi-commutators of quasi-homogeneous Toeplitz operators’, Complex Anal. Oper. Theory 2 (2008), 429–439.CrossRefGoogle Scholar
Čučković, Ž. and Rao, N. V., ‘Mellin transform, monomial symbols, and commuting Toeplitz operators’, J. Funct. Anal. 154 (1998), 195–214.CrossRefGoogle Scholar
Dai, X., Dong, X. T. and Gao, Y. X., ‘A canonical form of complex skew-symmetric compact operators and applications to Toeplitz operators’, Illinois J. Math. 68 (2024), 245–280.CrossRefGoogle Scholar
Ding, X., ‘The finite sum of the products of two Toeplitz operators’, J. Aust. Math. Soc. 86 (2009), 45–60.CrossRefGoogle Scholar
Ding, X., Qin, Y. and Zheng, D., ‘A theorem of Brown–Halmos type on the Bergman space modulo finite rank operators’, J. Funct. Anal. 273 (2017), 2815–2845.CrossRefGoogle Scholar
Ding, X. and Zheng, D., ‘Finite rank commutator of Toeplitz operators or Hankel operators’, Houston J. Math. 34 (2008), 1099–1119.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Guo, K., Sun, S. and Zheng, D., ‘Finite rank commutators and semi-commutators of Toeplitz operators with harmonic symbols’, Illinois J. Math. 51 (2007), 583–596.CrossRefGoogle Scholar
Le, T. and Thilakarathna, D., ‘Finite rank perturbations of Toeplitz products on the Bergman space’, J. Funct. Anal. 280 (2021), 108850.CrossRefGoogle Scholar
Le, T. and Thilakarathna, D., ‘A generalization of the Brown–Halmos theorems for the unit ball’, Adv. Math. 404 (2022), 108411.CrossRefGoogle Scholar
Louhichi, I. and Rao, N. V., ‘Bicommutants of Toeplitz operators’, Arch. Math. 91 (2008), 256–264.CrossRefGoogle Scholar
Louhichi, I., Strouse, E. and Zakariasy, L., ‘Products of Toeplitz operators on the Bergman space’, Integral Equations Operator Theory 54 (2006), 525–539.CrossRefGoogle Scholar
Louhichi, I. and Zakariasy, L., ‘On Toeplitz operators with quasihomogeneous symbols’, Arch. Math. 85 (2005), 248–257.CrossRefGoogle Scholar
Remmert, R., Classical Topics in Complex Function Theory, Graduate Texts in Methematics (Springer, New York, 1998).CrossRefGoogle Scholar
Stroethoff, K., The Berezin Transform and Operators on Spaces of Analytic Functions, Banach Center Publications, 38 (Polish Academy of Sciences, Warsaw, 1997), pp. 361–380.Google Scholar
Vasilevski, N., ‘Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators’, Integral Equations Operator Theory 66 (2010), 141–152.CrossRefGoogle Scholar
Zheng, D., ‘Commuting Toeplitz operators with pluriharmonic symbols’, Trans. Amer. Math. Soc. 350 (1998), 1595–1618.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar