Toeplitz quantization on Fock space

Abstract

For Toeplitz operators Tf(t) acting on the weighted Fock space Ht2, we consider the semi-commutator Tf(t)Tg(t)−Tfg(t), where t>0 is a certain weight parameter that may be interpreted as Planck's constant ħ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit(⁎)limt→0⁡‖Tf(t)Tg(t)−Tfg(t)‖t. It is well-known that ‖Tf(t)Tg(t)−Tfg(t)‖t tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to f,g∈BUC(Cn) by Bauer and Coburn. We now further generalize (⁎) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra VMO∩L∞ of bounded functions having vanishing mean oscillation on Cn. Our approach is based on the algebraic identity Tf(t)Tg(t)−Tfg(t)=−(Hf¯(t))⁎Hg(t), where Hg(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 limsupt→0‖Hg(t)‖t<∞, e.g. g∈L∞(Cn). We prove that the set of all symbols f∈L∞(Cn) with the property that limt→0⁡‖Tf(t)Tg(t)−Tfg(t)‖t=limt→0⁡‖Tg(t)Tf(t)−Tgf(t)‖t=0 for all g∈L∞(Cn) coincides with VMO∩L∞. Additionally, we show that limt→0⁡‖Tf(t)‖t=‖f‖∞ holds for all f∈L∞(Cn). Finally, we present new examples, including bounded smooth functions, where (⁎) does not vanish.