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 (TfTg(t))-T-(t) - T-fg((t)), where t > 0 is a certain weight parameter that may be interpreted as Planck's constant h in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit lim(t -> 0) parallel to(TfTg(t))-T-(t) - T-fg((t))parallel to t. It is well-known that parallel to(TfTg(t))-T-(t) - T-fg((t))parallel to t tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to f, g is an element of BUC(C-n) by Bauer and Coburn. We now further generalize (*) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra VMO boolean AND L-infinity of bounded functions having vanishing mean oscillation on C-n Our approach is based on the algebraic identity (TfTg(t))-T-(t) - T-fg((t)) = -(H(t)/f)*H-g((t)), where Hg(t) denotes the Henkel 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 -> 0) parallel to H-g((t))parallel to(t) < infinity, e.g. g is an element of L-infinity(C-n). We prove that the set of all symbols f is an element of L infinity(C-n) with the property that lim(t -> 0)parallel to(TfTg(t))-T-(t) - T-fg((t))parallel to(t) =lim(t -> 0) parallel to(TgTf(t))-T-(t) - T-gf((t))parallel to(t) for all g is an element of L-infinity(C-n) coincides with VMO boolean AND L-infinity. Additionally, we show that lim(t -> 0)parallel to T-f((t))parallel to(t) = parallel to f parallel to(infinity) holds for all f is an element of L-infinity(C-n). Finally, we present new examples, including bounded smooth functions, where (*) does not vanish. (C) 2018 Elsevier Inc. All rights reserved.