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 $α>0$ we denote by $B_α$ the Berezin transform associated to the measure $μ_{m}^α$ with density proportional to $e^{-α|z|^m}$ with respect to Lebesgue measure on the complex plane and normalized so that $μ_ϕ^α(\mathbb C)=1$. We show that the commutativity relation $B_αB_βf=B_βB_αf$ holds for all $f\in L^{\infty}(\mathbb C)$ and $α,β> 0$ if and only if $m=2$.