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For any function $f$ in $L^{\infty}(\mathbb{D})$, let $T_f$ denote thecorresponding Toeplitz operator the Bergman space $A^2(\mathbb{D})$. A recentresult of D. Luecking shows that if $T_f$ has finite rank then $f$ must be thezero function. Using a refined version of this result, we show that if allexcept 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.

A refined Luecking's theorem and finite-rank products of Toeplitz operators