  In this paper, we extend the Brown-Halmos theorems to the Fock space and
investigate the range of the Berezin transform. We observe that there are
non-pluriharmonic functions $u$ that can be written as a finite sum
$B(u)=\sum_lf_l\overline{g_l}$, where $f_l,g_l$ are holomorphic functions
belonging to the class $\mathrm{Sym}(\mathbb{C}^n)$. In addition, we solve an
open question about the zero product of Toeplitz operators, which was posed by
Bauer et al. in 2015. Our results reveal that the Brown-Halmos theorems on the
Fock space are more complicated than that on the classical Bergman space.
