On the Commutativity of the Berezin Transform

2025/10/09 by Alexander Borichev, Borichev, Alexander, Gérard Fantolini +3
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Mathematical Analysis and Transform Methods #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2510.07887

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∈ L^∞(\mathbb C) and α,β> 0 if and only if m=2.

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