Positivity of Toeplitz operators on harmonic Bergman space

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• 作者：Yong Lu Shu ; Xian Feng Zhao
• 关键词：Positive Toeplitz operators ; harmonic Bergman space ; Berezin transform
• 刊名：Acta Mathematica Sinica
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：32
• 期：2
• 页码：175-186
• 全文大小：207 KB
• 参考文献：[1]Axler, S.: Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., 171, Longman Sci. Tech., Harlow, (1988), 1–50
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  [4]Choe, B., Nam, K. K.: Berezin transform and Toeplitz operators on harmonic Bergman spaces. J. Funct. Anal., 257, 3135–3166 (2009)MATH MathSciNet CrossRef
  [5]Fine, H.: A College Algebra, reprint, Forgotten Books, London, 2013
  [6]Ramey, W., Yi, H.: Harmonic Bergman functions on half-spaces. Trans. Amer. Math. Soc., 348, 633–660 (1996)MATH MathSciNet CrossRef
  [7]Stroethoff, K.: Compact Toeplitz operators on weighted harmonic Bergman spaces. J. Austr. Math. Soc. (Ser. A), 64, 136–148 (1998)MATH MathSciNet CrossRef
  [8]Zhao, X., Zheng, D.: Positivity of Toeplitz operators via Berezin transform. J. Math. Anal. Appl., 416, 881–900 (2014)MATH MathSciNet CrossRef
  [9]Zhu, K.: Operator Theory in Function Spaces, 2nd ed, Mathematical Surveys and Monographs, 138, American Mathematical Society, Basel, New York, 2007CrossRef
  
• 作者单位：Yong Lu Shu (1)
  Xian Feng Zhao (1) (2)
  
  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China
  2. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Chinese Library of Science
  
• 出版者：Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, co-published
• ISSN：1439-7617

文摘

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.