  Let $m \geq 1$ be an integer and let $H_m(\mathbb B)$ be the analytic
functional Hilbert space on the unit ball $\mathbb B \subset \mathbb C^n$ given
by the reproducing kernel $K_m(z,w) = (1 - \langle z,w \rangle)^{-m}$. We prove
that Toeplitz operators with pluriharmonic symbol on $H_m(\mathbb B)$ can be
characterized by an algebraic identity which extends the classical Brown-Halmos
characterization of Toeplitz operators on the Hardy space of the unit disc as
well as corresponding results of Louhichi and Olofsson for Toeplitz operators
with harmonic symbol on weighted Bergman spaces of the disc.
