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A Lipschitz estimate for Berezin’s operator calculus. (English) Zbl 1093.47024
The author establishes some properties of the Berezin symbol $\widetilde{X}$ of a bounded operator $X$ acting in one of the following reproducing kernel Hilbert spaces: the Segal--Bargmann space of all holomorphic functions on ${\Bbb C}^n$ which are square integrable with respect to a Gaussian measure, or the Bergman space of all holomorphic functions on a bounded domain $ \Omega \subset {\Bbb C}^n$ which are square integrable with respect to Lebesgue measure. For the Segal--Bargmann space, it is shown that $\widetilde{X}$ is Lipschitz in its domain ${\Bbb C}^n$ with respect to the usual Euclidean distance. For the Bergman space, it is shown that $\widetilde{X}$ is Lipschitz in its domain $\Omega$, but now with respect to a distance defined in $\Omega$ via the reproducing kernel function. These are the main results of the article. In both cases, the Lipschitz constant of $\widetilde{X}$ is shown to be bounded above by $\sqrt{2}\Vert X \Vert$, where $\Vert X \Vert$ is the operator norm of $X$. However, no statement is made about the set of operators for which this bound is optimal. In the last section, two further results are proved for the Segal--Bargmann space. First, the function space of Berezin symbols is shown to be invariant under translations in ${\Bbb C}^n$. Next, it is shown that there is no bounded operator $X$ whose Berezin symbol satisfies $\widetilde{X}(a)= e^{-2\vert a\vert ^2}$ for all $a \in {\Bbb C}^n$, even though this is a Lipschitz function which satisfies all the other necessary conditions (as given in the article) to be a Berezin symbol.

47B32Operators in reproducing-kernel Hilbert spaces
32A36Bergman spaces
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