The eigenvalues of limits of radial Toeplitz operators

Let $A^2$ be the Bergman space on the unit disk. A bounded operator $S$ on $A^2$ is called radial if $S{z}^n={\lambda }_n{z}^n$ for all $n\ge 0$, where ${\lambda }_n$ is a bounded sequence of complex numbers. We characterize the eigenvalues of radial operators that can be approximated by Toeplitz operators with bounded symbols.