Finite Rank Perturbations of Toeplitz Products on the Bergman Space

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution $\diamond$ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if $F_j, G_j$ ($1\leq j\leq N$) are polynomials of $z$ and $\bar{z}$ then $\sum_{j=1}^{N}T_{F_j}T_{G_j}-T_{H}$ is a finite rank operator for some $L^{1}$-function $H$ if and only if $\sum_{j=1}^{N}F_j\diamond G_j$ belongs to $L^1$ and $H=\sum_{j=1}^{N}F_j\diamond G_j$. In the case $F_j$'s are holomorphic and $G_j$'s are conjugate holomorphic, it is shown that $H$ is a solution to a system of first order partial differential equations with a constraint.