Heat flow, BMO, and the compactness of Toeplitz operators

Abstract

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 < ∞ whenever g ˜ ( s ) ∈ C 0 ( C n ) (vanishes at infinity) or g ˜ ( s ) ∈ L p ( C n , d v ) , respectively, for some s with 0 < s < 1 4 , where g ˜ ( s ) is the heat transform of g on C n . Moreover, we show that compactness of T g implies that g ˜ ( s ) is in C 0 ( C n ) for all s > 1 4 and use this to show that, for g ∈ BMO 1 ( C n ) , we have g ˜ ( s ) is in C 0 ( C n ) for some s > 0 only if g ˜ ( s ) is in C 0 ( C n ) for all s > 0 . This “backwards heat flow” result seems to be unknown for g ∈ BMO 1 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 L a 2 ( B n , d v α ) , where the “heat flow” g ˜ ( s ) is replaced by the Berezin transform B α ( g ) on L a 2 ( B n , d v α ) for α > − 1 .