  In this paper we generalize the classical theorems of Brown and Halmos about
algebraic properties of Toeplitz operators to Bergman spaces 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 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.
