On the commutant of operators of multiplication by univalent functions

Abstract

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