On the Berezin-Toeplitz calculus (Q2731928)

The author considers an interesting and important problem, whether the composition of two Berezin-Toeplitz operators \(T_{\varphi} T_{\psi}\) acting on the Segal-Bargmann (or Fock) space of Gaussian square-integrable entire functions on \({\mathbb C}^n\) is again a Berezin-Toeplitz operator \(T_{\varphi}\) for some symbol \(\varphi\). It is shown that for several interesting algebras of functions on \({\mathbb C}^n\) (smooth Bochner algebras) the answer is affirmative, and the function \(\varphi\) is a certain ``twisted'' associative product \(\varphi \diamond \psi\) of the functions \(\varphi\) and \(\psi\). At the same time the author gives an example of (unbounded) \(C^{\infty}\) radial function \(\varphi\) which generates the bounded Toeplitz operator \(T_{\varphi}\) and such that for any function \(\psi\) (in the considered class of symbols) \(T_{\varphi}T_{\varphi} \neq T_{\psi}\).