On the Commutativity of the Berezin Transform

Abstract

We consider the commutativity problem for the Berezin transform on weighted Fock spaces. Given a real number $m>0$, for every $\hat{I}\pm >0$ we denote by $B_{\hat{I}}\pm$ the Berezin transform associated to the measure $\hat{I}{¼}_m^{\hat{I}}\pm$ with density proportional to $e^{-\hat{I}\pm |z{|}^m}$ with respect to Lebesgue measure on the complex plane and normalized so that $\hat{I}{¼}_{\ddot{I}}{•}^{\hat{I}}\pm (\mathbb{C})=1$. We show that the commutativity relation $B_{\hat{I}}\pm B_{\hat{I}}^{2}f=B_{\hat{I}}^2{B}_{\hat{I}}\pm f$ holds for all $f\in L^{\infty }(\mathbb{C})$ and $\hat{I}\pm ,{\hat{I}}^2>0$ if and only if $m=2$.