Sharp Berezin Lipschitz estimates

F.A. Berezin introduced a general ``symbol calculus" for linear operators on reproducing kernel Hilbert spaces. For the Segal-Bargmann space $H^2({\text{\bf C}}^n,d\mu )$ of Gaussian square-integrable entire functions on complex $n$-space, ${\text{\bf C}}^n$, or for the Bergman spaces $A^2 (\Omega)$ of Euclidean volume square-integrable holomorphic functions on bounded domains $\Omega$ in ${\text{\bf C}}^n$, we show here that earlier Lipschitz estimates for Berezin symbols of arbitrary bounded operators are sharp.