Abstract
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
where \(\Vert \cdot \Vert _t\) 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. Borthwick, Perspectives on quantization. Contemporary mathematics, vol 214. AMS, Providence, pp 23–37, 1998; Coburn, Commun Math Phys 149:415–424, 1992) 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.
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To Professor Sergei Grudsky on the occasion of his $$60{\mathrm{th}}$$ 60 th birthday.
This note partly was written at the workshop “Analytic Function Spaces and Operators on Them” at Tsinghua Sanya International Mathematical Form (TSIMF). W. Bauer acknowledges support through TSIMF.
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Bauer, W., Coburn, L.A. Uniformly continuous functions and quantization on the Fock space. Bol. Soc. Mat. Mex. 22, 669–677 (2016). https://doi.org/10.1007/s40590-016-0108-8
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DOI: https://doi.org/10.1007/s40590-016-0108-8