Uniformly continuous functions and quantization on the Fock space. (English) Zbl 1356.47034
Summary: With a family \((\mu _t)_{t>0}\) of Gaussian probability measures, we consider the scale \((H_t^2)_{>0}\) of \(\mu_t\)-square integrable entire functions on \(\mathbb C^n\). Here, \(t\) plays the role of Planck’s constant. For \(f\) and \(g\) in the space \(\mathrm{BUC}(\mathbb C^n)\) of all bounded and uniformly continuous complex valued functions on \(\mathbb C^n\), we show the asymptotic composition formula
\[
\lim\limits_{t\downarrow 0}\| T_f^{(t)}T_g^{(t)}-T^{(t)}_{fg}\|_t=0,\tag{1}
\]
where \(\|\cdot\|_t\) denotes the norm in \(\mathcal L(H_t^2)\) and \(T_f^{(t)}\) is the Toeplitz operator with symbol \(f\). Different from previously known results (e.g., [D. Borthwick, Contemp. Math. 214, 23–37 (1998; Zbl 0903.58013); the second author, Commun. Math. Phys. 149, No. 2, 415–424 (1992; Zbl 0829.46056)]), neither differentiability nor compact support of the operator symbols is assumed. We provide an example which indicates that (1) in general fails for rapidly oscillating bounded symbols.
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 53D50 | Geometric quantization |
| 81S10 | Geometry and quantization, symplectic methods |
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