Finite rank product theorems for Toeplitz operators on the half-space

Abstract

On the harmonic Bergman space of the half space in ${\mathit{R}}^n$, we show that if the product of two or more Toeplitz operators with harmonic symbols that have certain boundary smoothness has finite rank, then one of the symbols must be identically 0. Our methods require the number of factors in the product to depend on the dimension $n$.