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 $Sz^n = \lambda_n z^n$ 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.