On the Commutativity of the Berezin Transform
Alexander Borichev, Gérard Fantolini, El-Hassan Youssfi
TL;DR
This work investigates when the Berezin transform $B_{\alpha,m}$ on weighted Fock spaces commutes, i.e., $B_{\alpha,m}B_{\beta,m}=B_{\beta,m}B_{\alpha,m}$, for measures $d\mu_{\alpha,m}(z)=\frac{m\alpha^{2/m}}{2\pi\Gamma(2/m)}e^{-\alpha|z|^m}dA(z)$. The authors develop a kernel-expansion framework using the Stieltjes moments to express $B_{\alpha,m}$ in terms of an entire function $S_{\alpha,m}$ and then reduce commutativity to moment-type equalities $U_{\alpha,\beta}(n)$. They show that the commutativity for all $\alpha,\beta>0$ and all $f\in L^{\infty}(\mathbb{C})$ occurs if and only if $m=2$, with a parallel equivalence via test functions $f_{\delta}$ and the value at $0$, and they provide a detailed analysis distinguishing the non-integer, even, and $m=2$ cases. The result identifies the classical Fock space as the unique setting with universal commutativity and relies on a delicate combination of kernel expansions and moment techniques.
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 $m=2$.