A Theorem of Nehari Type on Weighted Bergman Spaces of the Unit Ball

Abstract

This paper shows that if $S$ is a bounded linear operator acting on the weighted Bergman spaces ${A}_{\alpha }^{2}$ on the unit ball in ${\mathbb{C}}^{n}$ such that $S{T}_{{z}_{i}}={T}_{{\overline{z}}_{i}}S\text{\,}(i=1,\ldots ,n)$, where ${T}_{{z}_{i}}={z}_{i}f$ and ${T}_{{\overline{z}}_{i}}=P({\overline{z}}_{i}f)$; and where $P$ is the weighted Bergman projection, then $S$ must be a Hankel operator.