Products of Toeplitz Operators on a Vector Valued Bergman Space

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product $T_F{T}_{G^{\ast }}$ on the vector valued Bergman space $L_a^2({\mathbb{C}}^n)$, where $F$ and $G$ are matrix symbols with scalar valued Bergman space entries. The results generalize existing results in the scalar valued Bergman space case. We also characterize boundedness and invertibility of Toeplitz products $T_F{T}_{G^{\ast }}$ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space.