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Zbl 1094.47034
Englis, Miroslav; Zhang, Genkai
(Engliš, Miroslav)
On the derivatives of the Berezin transform. (English)
[J] Proc. Am. Math. Soc. 134, No. 8, 2285-2294 (2006). ISSN 0002-9939; ISSN 1088-6826/e

Authors' abstract: Improving upon a recent result of L.~A.\ Coburn and J.~Xia [see {\tt L.~A.\ Coburn}, Proc.\ Am.\ Math.\ Soc.\ 133, No.~1, 127--131 (2005; Zbl 1093.47024)], 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.
[Paşc Găvruţă (Timişoara)]

MSC 2000:: *47B32 Operators in reproducing-kernel Hilbert spaces
32A36 Bergman spaces
32M15 Symmetric spaces (analytic spaces)

Keywords: Bergman kernel; Segal--Bargmann space; Fock space; Berezin transform; bounded symmetric domain; invariant differential operator

Citations: Zbl 1093.47024

Cited in: Zbl 1218.47036 Zbl 1132.47019

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