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 ${ℂ}^n$ such that $S{T}_{z_i}=T_{{\overline{z}}_i}S\left(i=1,\dots ,n\right)$, where $T_{z_i}=z_{i}f$ and $T_{{\overline{z}}_i}=P\left({\overline{z}}_{i}f\right)$; and where $P$ is the weighted Bergman projection, then $S$ must be a Hankel operator.