Products of Toeplitz operators on the Bergman spaces $A_\alpha^2$

We give a sufficient and a necessary condition for the product of Toeplitz operators $T^\alpha_fT^\alpha_{\bar g}$, with $f$, $g$ analytic, to be bounded on the weighted Bergman space $L^2_a(\mathbb D,(1-|z|^2)^\alpha dA)$. We also show that the only compact product of weighted Toeplitz operators is the trivial one.