On higher dimensional Luecking's theorem (Q1006185)

Let \(\Omega\) be a complete circular domain in \(\mathbb{C}^n\), and let \(\mu\) be a complex Borel measure on \(\Omega\). The main result of the paper states that the Toeplitz operator \(T_{\mu}\) acting on the Bergman space of analytic (or pluriharmonic) functions in \(\Omega\) has a finite rank if and only if \(\mu\) is a linear combination of point masses. This result provides to a higher-dimensional extension of a result of \textit{D.\,H.\thinspace Luecking} [Proc.\ Am.\ Math.\ Soc.\ 136, No.\,5, 1717--1723 (2008; Zbl 1152.47021)].