On the Commutativity of the Berezin Transform

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