Positivity of Toeplitz Operators on Harmonic Bergman Space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative o...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2016-02, Vol.32 (2), p.175-186
Main Authors: Shu, Yong Lu, Zhao, Xian Feng
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Berezin变换
Disks
Functions (mathematics)
Half planes
Harmonics
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Operators
Studies
Symbols
Toeplitz算子
Transforms
上半平面
可调和
极性
磁盘
空间算子
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Summary:In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-016-5138-7