Heat Flow and An Algebra of Toeplitz Operators

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• 作者：Dieudonne Agbor ; Wolfram Bauer
• 关键词：Primary 47B35 ; Secondary 53D55 ; Fock space ; composition formulas ; Berezin–Toeplitz quantization ; * ; product in deformation quantization ; Berezin transform
• 刊名：Integral Equations and Operator Theory
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：81
• 期：2
• 页码：271-299
• 全文大小：389 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8989

文摘

We define a family of associative products \({(\sharp_s)_{s>0}}\) on a space S ?/sub> of real analytic functions on \({\mathbb{C}^n}\) that are contained in the range of the heat transform for all times t?>?0. Extending results in Bauer (J Funct Anal 256:3107-142, 2009), Coburn (J Funct Anal 161:509-25, 1999; Proc Am Math Soc 129(11):3331-338, 2007) we show that this product leads to composition formulas of in general unbounded Berezin–Toeplitz operators \({T^{(s)}_f}\) on \({H_s^2}\) having symbols \({f \in S_{\infty}}\) . Here \({H_s^2}\) denotes the Segal–Bargmann space over \({\mathbb{C}^n}\) with respect to the semi-classical parameter s?>?0. In the special case of operators with polynomial symbols or for products of just two operators such formulas previously have been obtained in Bauer (J Funct Anal 256:3107-142, 2009), Coburn (Proc Am Math Soc 129(11):3331-338, 2007), respectively. Finally we give an example of a bounded real analytic function h on \({\mathbb{C}}\) such that \({(T_h^{(1)})^2}\) cannot be expressed in form of a Toeplitz operator \({T_g^{(1)}}\) where g fulfills a certain growth condition at infinity.