Abstract. The setting of this talk is the weighted Fock space $F^2_m$ of entire functions on $\mathbb{C}$ which are square integrable with respect to the measure $d\mu_m(z)=e^{-|z|^{2m}}dz$, $m>0$, where $dz$ is the normalized Lebesgue measure.

In the context of the Segal-Bargmann space  ($m=1$), Cho, Park and Zhu studied the boundedness of the product of Toeplitz operators $T_u T_{\overline{v}}$ on $F^2_1$.

We extend their work to the case of general $m\geq 1$, and give necessary and sufficient conditions on $u$, $v$ in $F^2_m$ for the product $T_u T_{\overline{v}}$ to be bounded on $F^2_m$. In particular, we relate the boundedness of $T_u T_{\overline{v}}$ with the boundedness of product of the Berezin transforms of $|u|^2$ and $|v|^2$ (Sarason's conjecture).

Joint work with E. H. Youssfi and K. Zhu.

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