Hilbert-Schmidt Hankel operators with harmonic symbols on the Bergman space of strongly pseudoconvex domains in $\mathbb{C}^n$

TL;DR

This paper characterizes when Hankel operators with harmonic symbols are Hilbert-Schmidt on Bergman spaces of strongly pseudoconvex domains in complex spaces, showing they are Hilbert-Schmidt if and only if the symbols are holomorphic.

Contribution

It provides a complete characterization of Hilbert-Schmidt Hankel operators with harmonic symbols on Bergman spaces of strongly pseudoconvex domains in several complex variables.

Findings

Hankel operator $H_{}$ is Hilbert-Schmidt iff $$ is holomorphic.

Harmonic symbols of class $C^3$ are considered.

The characterization holds for domains in $C^n$, $n geq 2$.

Abstract

We characterize Hilbert-Schmidt Hankel operators on the Bergman spaces of smooth bounded strongly pseudoconvex domains in $\mathbb{C}^n$ for $n \geq 2$. We consider harmonic symbols of class $C^3$ up to the closure of the domain and show $H_{\phi}$ is Hilbert-Schmidt if and only if $\phi$ is holomorphic on the domain.