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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 X ˜ of a bounded operator X acting either in the Segal-Bargmann space H 2 ( n ,dμ) or in the Bergman space A 2 (Ω). 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:
47B32Operators in reproducing-kernel Hilbert spaces
32A36Bergman spaces