## Toeplitz quantization on Fock space
Bauer, W., Coburn, L. A. and Hagger, R.
(2018)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1016/j.jfa.2018.01.001 ## Abstract/SummaryFor 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 $\hbar$ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit \begin{equation}\tag{$*$} \lim\limits_{t \to 0} \|T_f^{(t)}T_g^{(t)} - T_{fg}^{(t)}\|_t. \end{equation} It is well-known that $\|T_f^{(t)}T_g^{(t)} - T_{fg}^{(t)}\|_t$ tends to $0$ under certain smoothness assumptions imposed on $f$ and $g$. This result was recently extended to $f,g \in \textup{BUC}(\C^n)$ by Bauer and Coburn. We now further generalize $(*)$ to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\textup{VMO}\cap L^{\infty}$ of bounded functions having vanishing mean oscillation on $\mathbb{C}^n$. Our approach is based on the algebraic identity $T_f^{(t)}T_g^{(t)} - T_{fg}^{(t)} = -(H_{\bar{f}}^{(t)})^*H_g^{(t)}$, 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 $(*)$ to vanish. For $g$ we only have to impose $\limsup_{t \to 0} \|H_g^{(t)}\|_t < \infty$, e.g.~$g \in L^{\infty}(\C^n)$. We prove that the set of all symbols $f\in L^{\infty} (\mathbb{C}^n)$ with the property that $\lim_{t \rightarrow 0}\|T^{(t)}_fT^{(t)}_g-T^{(t)}_{fg}\|_t= \lim_{t \rightarrow 0} \| T_g^{(t)}T_f^{(t)}-T_{gf}^{(t)}\|_t=0$ for all $g\in L^{\infty}(\mathbb{C}^n)$ coincides with $\textup{VMO}\cap L^{\infty}$. Additionally, we show that $\lim\limits_{t \to 0} \|T_f^{(t)}\|_t = \|f\|_{\infty}$ holds for all $f \in L^{\infty}(\C^n)$. Finally, we present new examples, including bounded smooth functions, where $(*)$ does not vanish.
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