Positivity of Toeplitz operators via Berezin transform |
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Authors: | Xianfeng Zhao Dechao Zheng |
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Affiliation: | 1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, PR China;2. Center of Mathematics, Chongqing University, Chongqing, 401331, PR China;3. Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States |
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Abstract: | In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of |z| on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive. |
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Keywords: | Bergman space Positive Toeplitz operators Berezin transform |
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