Compact Hankel operators on generalized Bergman spaces of the polydisc

TL;DR

This paper characterizes the compactness of Hankel operators on generalized Bergman spaces of the polydisc, linking it to boundary behavior and function decomposition.

Contribution

It provides a necessary and sufficient condition for the compactness of Hankel operators on these spaces, involving a specific function decomposition.

Findings

Hankel operator $H_f$ is compact iff $f=h+g$ with $h$ in the ball algebra and $g$ vanishing on the boundary.

The result extends understanding of operator compactness in multivariable complex analysis.

Connects boundary behavior of functions to operator properties on Bergman spaces.

Abstract

We show that for $f$ a continuous function on the closed polydisc $\bar{\mathbb{D}^n}$ with $n\geq 2$, the Hankel operator $H_{f}$ is compact on the Bergman space of $\mathbb{D}^n$ if and only if there is a decomposition $f=h+g$, where $h$ is in the ball algebra and $g$ vanishes on the boundary of the polydisc.