Bulletin of the
Korean Mathematical Society BKMS

Let ${\mathcal B}$ be a certain Banach space consisting of analytic functions defined on a bounded domain $G$ in the complex plane. Let $\varphi$ be an analytic polynomial or a rational function and let $M_{\varphi}$ denote the operator of multiplication by $\varphi$. Under certain condition on $\varphi$ and $G$, we characterize the commutant of $M_{\varphi}$ that is the set of all bounded operators $T$ such that $TM_{\varphi}=M_{\varphi}T$. We show that $T=M_{\Psi}$ for some function $\Psi$ in ${\mathcal B}$.

Keywords: commutant, multiplication operators, Banach space of analytic functions, univalent function, bounded point evaluation.

MSC numbers: Primary 47B35; Secondary 47B38