A Lipschitz estimate for Berezin’s operator calculus.

*(English)*Zbl 1093.47024The 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 \({\mathbb C}^n\) which are square integrable with respect to a Gaussian measure, or the Bergman space of all holomorphic functions on a bounded domain \( \Omega \subset {\mathbb C}^n\) 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 \({\mathbb C}^n\) 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 \(\sqrt{2}\| X \|\), where \(\| X \|\) 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 \({\mathbb C}^n\). Next, it is shown that there is no bounded operator \(X\) whose Berezin symbol satisfies \(\widetilde{X}(a)= e^{-2| a| ^2}\) for all \(a \in {\mathbb C}^n\), even though this is a Lipschitz function which satisfies all the other necessary conditions (as given in the article) to be a Berezin symbol.

For the Segal–Bargmann space, it is shown that \(\widetilde{X}\) is Lipschitz in its domain \({\mathbb C}^n\) 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 \(\sqrt{2}\| X \|\), where \(\| X \|\) 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 \({\mathbb C}^n\). Next, it is shown that there is no bounded operator \(X\) whose Berezin symbol satisfies \(\widetilde{X}(a)= e^{-2| a| ^2}\) for all \(a \in {\mathbb C}^n\), even though this is a Lipschitz function which satisfies all the other necessary conditions (as given in the article) to be a Berezin symbol.

Reviewer: Stephen B. Sontz (Guanajuato)