m$-Berezin transform and compact operators

$m$-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.