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FINITE LINEAR COMBINATIONS OF MONOMIAL TOEPLITZ PRODUCTS ON THE BERGMAN SPACE

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

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).

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