On the derivatives of the Berezin transform

Engliš, Miroslav; Zhang, Genkai

doi:10.1090/S0002-9939-06-08238-4

On the derivatives of the Berezin transform
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by Miroslav Engliš and Genkai Zhang PDF

Proc. Amer. Math. Soc. 134 (2006), 2285-2294 Request permission

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.

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