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On the Berezin-Toeplitz calculus

Authors:	L A Coburn

Institution:	Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260

Abstract:	We consider the problem of composing Berezin-Toeplitz operators on the Hilbert space of Gaussian square-integrable entire functions on complex -space, $\mathbf{C}^n$ . For several interesting algebras of functions on $\mathbf{C}^n$ , we have $T_\varphi T_\psi =T_{\varphi \diamond \psi }$ for all $\varphi ,\psi$ in the algebra, where $T_\varphi$ is the Berezin-Toeplitz operator associated with $\varphi$ and $\varphi \diamond \psi$ is a ``twisted' associative product on the algebra of functions. On the other hand, there is a $C^\infty$ function $\varphi$ for which $T_\varphi$ is bounded but $T_\varphi T_\varphi \neq T_\psi$ for any $\psi$ .

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