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 $e^{-\alpha \mathrm{\mid }z{\mathrm{\mid }}^m}$ 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 $f\in L^{\mathrm{\infty }}(\mathbb{C})$ and $\alpha ,\beta >0$ if and only if $m=2$.