On the commutant of operators of multiplication by univalent functions

Let $\mathcal{B}$ be a certain Banach space consisting of continuous functions defined on the open unit disk. Let ${\phi}\in \mathcal{B}$ be a univalent function defined on $\overline{\mathbf{D}}$, and assume that $M_{\phi}$ denotes the operator of multiplication by ${\phi}$. We characterize the structure of the operator $T$ such that $M_{\phi} T=T M_{\phi}$. We show that $T=M_{\varphi}$ for some function ${\varphi}$ in $\mathcal{B}$. We also characterize the commutant of $M_{{\phi}^2}$ under certain conditions.