$m$-Berezin transforms are introduced for bounded operators on the Bergman
space of the unit ball. The norm of the $m$-Berezin transform as a linear
operator from the space of bounded operators to $L^{\infty}$ is found. We
show that the $m$-Berezin transforms are commuting with each other and
Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball.
Using the $m$-Berezin transforms we show that a radial operator in the
Toeplitz algebra is compact iff its Berezin transform vanishes on the
boundary of the unit ball.