Products of Toeplitz operators on the Fock space

Abstract

Let $f$ and $g$ be functions, not identically zero, in the Fock space $F^2$ of $C_n$. We show that the product $T_f{T}_{\bar{g}}$ of Toeplitz operators on $F^2$ is bounded if and only if $f(z)=e^{q(z)}$ and $g(z)=c{e}^{-q(z)}$, where $c$ is a nonzero constant and $q$ is a linear polynomial.