  For any function $f$ in $L^{\infty}(\mathbb{D})$, let $T_f$ denote the
corresponding Toeplitz operator the Bergman space $A^2(\mathbb{D})$. A recent
result of D. Luecking shows that if $T_f$ has finite rank then $f$ must be the
zero function. Using a refined version of this result, we show that if all
except possibly one of the functions $f_1,..., f_{m}$ are radial and
$T_{f_1}... T_{f_m}$ has finite rank, then one of these functions must be zero.
