Heat flow, BMO, and the compactness of Toeplitz operators

In this paper, it is shown that the Berezin-Toeplitz operator T-g is compact or in the Schatten class S-p of the Segal-Bargmann space for 1 <= p < infinity whenever <(g)over tilde>((s)) is an element of C-0 (C-n) (vanishes at infinity) or (g) over tilde)((s)) is an element of L-p (C-n, dv), respectively, for some s with 0 <s < 1/4, where (g) over tilde ((s)) is the heat transform of g on C-n. Moreover, we show that compactness of T-g implies that (g) over tilde ((s)) is in Co(C-n) for all s > 1/4 and use this to show that, for g is an element of BMO1 (C-n), we have (g) over tilde ((s)) is in C-0(C-n) for some s > 0 only if (g) over tilde ((s)) is in C-0(C-n) for all s > 0. This "backwards heat flow" result seems to be unknown for g is an element of BMO1 and even g is an element of L-infinity. Finally, we show that our compactness and vanishing "backwards heat flow" results hold in the context of the weighted Bergman space L-a(2) (B-n, dv(alpha)), where the "heat flow" (g) over tilde ((s)) is replaced by the Berezin transform B-alpha(g) on L-a(2) (B-n, dv(alpha)) for alpha > -1. (C) 2010 Elsevier Inc. All rights reserved.