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 all s>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.