In this paper we generalize the classical theorems of Brown and Halmos about
algebraic properties of Toeplitz operators to the Bergman space over the unit
ball in several complex variables. A key result, which is of independent
interest, is the characterization of summable functions $u$ on the unit ball
whose Berezin transform $B(u)$ can be written as a finite sum
$\sum_{j}f_j\,\bar{g}_j$ with all $f_j, g_j$ being holomorphic. In particular,
we show that such a function must be pluriharmonic if it is sufficiently smooth
and bounded. We also settle an open question about $\mathcal{M}$-harmonic
functions. Our proofs employ techniques and results from function and operator
theory as well as partial differential equations.
