Compactness of products of Hankel operators on convex Reinhardt domains in C^2

TL;DR

This paper investigates the conditions under which the product of Hankel operators on convex Reinhardt domains in C^2 is compact, revealing that either symbol must be holomorphic on boundary disks if the product is compact.

Contribution

It establishes a boundary holomorphicity condition for symbols of Hankel operators whose product is compact on convex Reinhardt domains in C^2.

Findings

If the product of Hankel operators is compact, then on each boundary disk, either f or g is holomorphic.

The result links operator compactness to boundary regularity and holomorphicity of symbols.

Provides a characterization of compactness for Hankel operator products in several complex variables.

Abstract

Let D be a piecewise smooth bounded convex Reinhardt domain in C^2. Assume that the symbols f and g are continuous on the closure of D and harmonic on the disks in the boundary of D. We show that if the product of Hankel operators H^*_f H_g is compact on the Bergman space of D, then on any disk in the boundary of D, either f or g is holomorphic.