Hilbert-Schmidt Hankel operators over semi-Reinhardt domains

TL;DR

This paper proves that for semi-Reinhardt domains in complex space, any Hilbert-Schmidt Hankel operator with an anti-holomorphic symbol must be zero when the domain dimension parameter m is at least 2, extending previous results from Reinhardt domains.

Contribution

It extends the known result that Hilbert-Schmidt Hankel operators with anti-holomorphic symbols are zero from Reinhardt to semi-Reinhardt domains in complex analysis.

Findings

Hilbert-Schmidt Hankel operators with anti-holomorphic symbols are zero for m ≥ 2 on semi-Reinhardt domains.

The result generalizes previous findings from Reinhardt to semi-Reinhardt domains.

The proof applies to arbitrary bounded semi-Reinhardt domains in complex space.

Abstract

Let $\Omega$ be an arbitrary bounded semi-Reinhardt domain in $\mathbb{C}^{m+n}$. We show that for $m \geq 2$, if a Hankel operator with an anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $L_a^2(\Omega)$, then it must equal zero. This fact has previously been proved for Reinhardt domains.