Abstract. Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator T on the Segal-Bargmann space, the Berezin transform of T is a function whose partial derivatives of all orders are bounded. Similarly, if T 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 T are bounded. Further generalizations are also discussed. 1
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