Heat Flow and An Algebra of Toeplitz Operators

Abstract

We define a family of associative products $${(\sharp_s)_{s>0}}$$ on a space S ∞ 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 256:3107–3142, 2009), Coburn (J Funct Anal 161:509–525, 1999; Proc Am Math Soc 129(11):3331–3338, 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 256:3107–3142, 2009), Coburn (Proc Am Math Soc 129(11):3331–3338, 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.