On operators which commute with analytic Toeplitz operators modulo the finite rank operators

It is shown that an operator $S$ on the Hardy space $H^2({\mathbb{D}}^n)$ (or $H^2({\mathbb{B}}_n)$) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if $S=T_g+F$. Here $F$ is a finite rank operator, and in the case $n=1$, $g$ is a sum of a rational function and a bounded analytic function, and in the case $n\geq 2$, $g$ is a bounded analytic function.