Products of Berezin Transforms on Generalized Fock Spaces

We study products $B_{\alpha ,m}^{(d)}{B}_{\beta ,m}^{(d)}$ of Berezin transforms on generalized Fock spaces over ${\mathbb{C}}^d$. We establish a result for $m\mathrm{\ne }2$: if $B_{\alpha ,m}^{(d)}{B}_{\beta ,m}^{(d)}$ coincides with another such product, then the weight parameters of the second product must be $(\alpha ,\beta )$. As consequences, we characterize when two Berezin transforms commute and when the product of two Berezin transforms is again a Berezin transform.