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Sharp Berezin Lipschitz estimates. (English) Zbl 1137.47018

The author in [Proc. Am. Math. Soc. 133, No. 1, 127–131 (2005; Zbl 1093.47024)] proved explicit Lipschitz estimates for the Berezin symbol $\stackrel{˜}{X}$ of a bounded operator $X$ acting either in the Segal-Bargmann space ${H}^{2}\left({ℂ}^{n},d\mu \right)$ or in the Bergman space ${A}^{2}\left({\Omega }\right)$. 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.

##### MSC:
 47B32 Operators in reproducing-kernel Hilbert spaces 32A36 Bergman spaces