Commutators and Semi-commutators of Toeplitz Operators on the Unit Ball

Monomial Toeplitz operators on the Bergman space and the pluriharmonic Bergman space of the unit ball in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb C}^n$$\end{document} are natural generalizations of the classical unilateral and bilateral shift operators, respectively. In this paper we completely characterize when the commutator and semi-commutator of two monomial Toeplitz operators on the unit ball have finite rank.