Finite-Rank Products of Toeplitz Operators in Several Complex Variables

Abstract.

For any α > −1, let A²_α be the weighted Bergman space on the unit ball corresponding to the weight (1 – |z|²)^α. We show that if all except possibly one of the Toeplitz operators $T_{f_{1} },\ldots,T_{f_{r}}$ are diagonal with respect to the standard orthonormal basis of A²_α and $T_{f_{1}} \cdots T_{f_{r}}$ has finite rank, then one of the functions $f_{1} ,\ldots, f_{r}$ must be the zero function.