On the commutant of operators of multiplication by univalent functions

Abstract:

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.