Products of Berezin Transforms on Generalized Fock Spaces

Abstract

We study products $B_{α,m}^{(d)}B_{β,m}^{(d)}$ of Berezin transforms on generalized Fock spaces over $\mathbb C^d$. We establish a result for $m\neq2$: if $B_{α,m}^{(d)}B_{β,m}^{(d)}$ coincides with another such product, then the weight parameters of the second product must be $(α,β)$. As consequences, we characterize when two Berezin transforms commute and when the product of two Berezin transforms is again a Berezin transform.