Toeplitz Quantization on Fock Space

For Toeplitz operators $T_f^{(t)}$ acting on the weighted Fock space $H_t^2$, we consider the semi-commutator $T_f^{(t)}{T}_g^{(t)}-T_{fg}^{(t)}$, where $t>0$ is a certain weight parameter that may be interpreted as Planck’s constant $\mathrm{\hslash }$ in Rieffel’s deformation quantization. In particular, we are interested in the semi-classical limit \tag{$\ast$}\lim\limits_{t\to 0}|T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}|t. It is well-known that $\mathrm{\parallel }{T}_f^{(t)}{T}_g^{(t)}-T_{fg}^{(t)}{\mathrm{\parallel }}_t$ tends to $0$ under certain smoothness assumptions imposed on $f$ and $g$. This result was extended to $f,g \in $\mathrm{B}\mathrm{U}\mathrm{C}$($\mathbb{C}$^n)$ in a recent paper by Bauer and Coburn. We now further generalize this result to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\mathrm{V}\mathrm{M}\mathrm{O}\cap L^{\mathrm{\infty }}$ of bounded functions having vanishing mean oscillation on ${\mathbb{C}}^n$. Our approach is based on the algebraic identity $T_f^{(t)}{T}_g^{(t)}-T_{fg}^{(t)}=-(H_{\bar{f}}^{(t)}{)}^{\ast }{H}_g^{(t)}$, where $H_g^{(t)}$ denotes the Hankel operator corresponding to the symbol $g$, and norm estimates in terms of the (weighted) heat transform. As a consequence, only $f$ (or likewise only $g$) has to be contained in one of the above classes for $(\ast )$ to vanish. For $g$ we only have to impose $\limsup{t \to 0}|H_g^{(t)}|t<\infty$,e\mathrm{.}g\mathrm{.}$g \in L^{\infty}($\mathbb{C}$^n)$. We prove that the set of all symbols $f\in L^{\mathrm{\infty }}({\mathbb{C}}^n)$ with the property that $\lim{t \rightarrow 0}|T^{(t)}fT^{(t)}g-T^{(t)}{fg}|t=\lim{t\to 0}|T_g^{(t)}T_f^{(t)}-T{gf}^{(t)}|_t=0$forall$g\in L^{\infty}($\mathbb{C}$^n)$coincideswith$$\mathrm{V}\mathrm{M}\mathrm{O}$\cap L^{\infty}$. Additionally, we show that ${\lim }_{t\to 0}\mathrm{\parallel }{T}_f^{(t)}{\mathrm{\parallel }}_t=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ holds for all $f\in L^{\mathrm{\infty }}({\mathbb{C}}^n)$. Finally, we present new examples, including bounded smooth functions, where $(\ast )$ does not vanish.