Heat flow, BMO, and the compactness of Toeplitz operators

In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1?p<∞ whenever (vanishes at infinity) or , respectively, for some s with , where is the heat transform of g on Cn. Moreover, we show that compactness of Tg implies that is in C₀(Cn) for all and use this to show that, for g∈BMO¹(Cn), we have is in C₀(Cn) for some s>0 only if is in C₀(Cn) for alls>0. This “backwards heat flow” result seems to be unknown for g∈BMO¹ and even g∈L^∞. Finally, we show that our compactness and vanishing “backwards heat flow” results hold in the context of the weighted Bergman space , where the “heat flow” is replaced by the Berezin transform Bα(g) on for α>−1.