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Zbl 1093.47024
Coburn, L.A.
A Lipschitz estimate for Berezin's operator calculus. (English)
[J] Proc. Am. Math. Soc. 133, No. 1, 127-131 (2005). ISSN 0002-9939; ISSN 1088-6826/e

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.
[Stephen B. Sontz (Guanajuato)]

MSC 2000:: *47B32 Operators in reproducing-kernel Hilbert spaces
32A36 Bergman spaces

Keywords: Berezin symbols; Berezin calculus; Lipschitz function; Segal-Bargmann space; Bergman space

Cited in: Zbl 1218.47036 Zbl 1241.47020 Zbl 1132.47019 Zbl 1137.47018 Zbl 1123.47001 Zbl 1125.47020 Zbl 1094.47034

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