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 ˜g(s) ∈ C0(Cn) (vanishes at infinity) or ˜g(s) ∈ Lp(Cn, dv), respectively, for some s with 0 < s < 14 , where ˜ g(s) is the heat transform of g on Cn.Moreover, we show that compactness of Tg implies that ˜g(s) is in C0(Cn) for alls > 14 and use this to show that, for g ∈ BMO1(Cn), we have ˜g(s) is in C0(Cn) for somes >0 only if ˜g(s) 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 L2a (Bn, dvα), where the “heat flow” ˜ g(s) is replaced by the Berezin transform Bα(g) on L2a (Bn, dvα) for α >−1. © 2010 Elsevier Inc. All rights reserved