On the derivatives of the Berezin transform

Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator on the Segal-Bargmann space, the Berezin transform of is a function whose partial derivatives of all orders are bounded. Similarly, if is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined ``invariant derivatives' of any order of the Berezin transform of are bounded. Further generalizations are also discussed.