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- W4302776630 abstract "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 $hbar$ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit tag{$*$}limlimits_{tto 0}|T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}|_t. 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 extended to $f,g in mathrm{BUC}(mathbb{C}^n)$ 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 $mathrm{VMO} cap L^{infty}$ 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. $g in L^{infty}(mathbb{C}^n)$. We prove that the set of all symbols $fin L^{infty}(mathbb{C}^n)$ with the property that $lim_{t rightarrow 0}|T^{(t)}_fT^{(t)}_g-T^{(t)}_{fg}|_t=lim_{tto 0}|T_g^{(t)}T_f^{(t)}-T_{gf}^{(t)}|_t=0$ for all $gin L^{infty}(mathbb{C}^n)$ coincides with $mathrm{VMO}cap L^{infty}$. Additionally, we show that $lim_{tto 0}|T_f^{(t)}|_t=|f|_{infty}$ holds for all $fin L^{infty}(mathbb{C}^n)$. Finally, we present new examples, including bounded smooth functions, where $(*)$ does not vanish." @default.
- W4302776630 created "2022-10-07" @default.
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- W4302776630 date "2017-04-19" @default.
- W4302776630 modified "2023-10-18" @default.
- W4302776630 title "Toeplitz Quantization on Fock Space" @default.
- W4302776630 doi "https://doi.org/10.48550/arxiv.1704.05652" @default.
- W4302776630 hasPublicationYear "2017" @default.
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