Zero products of Toeplitz operators on Reinhardt domains

TL;DR

This work addresses the zero-product problem for finitely many Toeplitz operators on the Bergman space $A^2(\Omega)$ of a bounded Reinhardt domain, with symbols that are finite sums of bounded quasi-homogeneous functions. The authors show that if $T_{\phi_m}\cdots T_{\phi_1}=0$, then at least one symbol must vanish, extending known results beyond the unit disk and ball to more general Reinhardt domains. The proof combines a Fourier (angular) decomposition of quasi-homogeneous symbols with an induction on the dimension of index supports, using a (I)-type condition on zero-sets to force vanishing. A corollary handles cases with an $L^{\infty}$ symbol, emphasizing the rigidity of zero-product phenomena for bounded domains. The work advances understanding of algebraic properties of Toeplitz operators on complex domains and suggests new tools for broader settings.

Abstract

Let $Ω$ be a bounded Reinhardt domain in $\mathbb{C}^n$ and $φ_1,\ldots,φ_m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{φ_m}\cdots T_{φ_1}=0$ on the Bergman space on $Ω$, then $φ_j=0$ for some $j$.

Theorems & Definitions (13)

• Theorem 1
• Corollary 1
• Definition 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 2
• Proposition 1