Powers and roots of Toeplitz operators

We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We use this to characterize Toeplitz operators with symbols in $L^{\infty}(\mathbb{D})$ which commute with Toeplitz operators whose symbols are of the form $e^{ip\theta}r^m$.