Commutators and Semi-commutators of Monomial Toeplitz Operators on the Pluriharmonic Hardy Space

In this paper, the authors completely characterize the finite rank commutator and semi-commutator of two monomial Toeplitz operators on the pluriharmonic Hardy space of the torus or the unit sphere. As a consequence, many non-trivial examples of (semi-)commuting Toeplitz operators on the pluriharmonic Hardy spaces are given.