Zero products of Toeplitz operators on Reinhardt domains

TL;DR

This paper proves that on bounded Reinhardt domains, the vanishing product of Toeplitz operators with quasi-homogeneous symbols implies at least one of the symbols must be zero, revealing a rigidity property of these operators.

Contribution

It establishes a new zero-product property for Toeplitz operators with quasi-homogeneous symbols on Reinhardt domains, extending understanding of their algebraic structure.

Findings

If the product of Toeplitz operators is zero, then some symbol must be zero.

The result applies to symbols that are finite sums of bounded quasi-homogeneous functions.

The theorem holds on bounded Reinhardt domains in complex space.

Abstract

Let $\Omega$ be a bounded Reinhardt domain in $\mathbb{C}^n$ and $\phi_1,\ldots,\phi_m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi_m}\cdots T_{\phi_1}=0$ on the Bergman space on $\Omega$, then $\phi_j=0$ for some $j$.