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Zbl 1137.47018
Coburn, L.A.
Sharp Berezin Lipschitz estimates.
(English)
[J] Proc. Am. Math. Soc. 135, No. 4, 1163-1168 (2007). ISSN 0002-9939; ISSN 1088-6826/e

The author in [Proc. Am. Math. Soc. 133, No.~1, 127--131 (2005; Zbl 1093.47024)] proved explicit Lipschitz estimates for the Berezin symbol $\widetilde{X}$ of a bounded operator $X$ acting either in the Segal-Bargmann space $H^2(\Bbb{C}^n, d\mu)$ or in the Bergman space $A^2(\Omega)$. By a careful choice of a family of operators $X_t$, where $t$ is a real parameter, it is shown here that these estimates are sharp, that is, the constants in them cannot be improved. Unfortunately, the motivation for constructing the family $X_t$ is not presented. Actually, $X_t$ is a rank two selfadjoint operator for each value of $t$. There is no discussion of whether these estimates could be shown to be sharp using just one operator $X$. It would also be interesting to know if this is possible and, if so, to give a characterization of such operators~$X$. The article concludes with two similar open problems. On a touching note, all too rare in the scientific literature, the author dedicates the article to the memory of his late wife.
[Stephen B. Sontz (Guanajuato)]
MSC 2000:
*47B32 Operators in reproducing-kernel Hilbert spaces
32A36 Bergman spaces

Keywords: Lipschitz estimates; Berezin symbol; Segal-Bargmann space; Bergman space

Citations: Zbl 1093.47024

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