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Positivity of Toeplitz operators on harmonic Bergman space

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Abstract

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.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Yong Lu Shu & Xian Feng Zhao

  2. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Xian Feng Zhao

Authors
  1. Yong Lu Shu
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  2. Xian Feng Zhao
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Corresponding author

Correspondence to Xian Feng Zhao.

Additional information

Supported by NSFC (Grant No. 11271387) and Chongqing Natural Sience Foundation (Grant No. cstc 2013jjB0050)

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Shu, Y.L., Zhao, X.F. Positivity of Toeplitz operators on harmonic Bergman space. Acta. Math. Sin.-English Ser. 32, 175–186 (2016). https://doi.org/10.1007/s10114-016-5138-7

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  • DOI: https://doi.org/10.1007/s10114-016-5138-7

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