A theorem of Brown-Halmos type on the Bergman space modulo finite rank operators

In this paper, we study the product problem of Toeplitz operators on the Bergman space of the unit disk. We characterize when the product of two Toeplitz operators TfTg is a finite rank perturbation of another Toeplitz operator Th, with f, g bounded harmonic and h in C2 class with invariant Laplacian in L1. As a consequence, we show that there is no nontrivial rank one perturbation. However, in the case rank m≥2, we construct an example that shows there are bounded harmonic functions f, g and h such that TfTg−Th has rank exactly m.