On the Essential Commutant of Toeplitz Operators in Several Complex Variables

Abstract.

Using the joint local mean oscillation, Jingbo Xia [13] showed that the essential commutant of $\mathfrak{T}(\mathcal{L})$, where ${\mathcal{L}}$ is the subalgebra of L ^∞ generated by all functions which are bounded and have at most one discontinuity, is $\mathfrak{T}$ (QC). Even though Xia’s method cannot be used, we are able to generalize his result to Toeplitz operators in higher dimensions with a different approach. This result is stronger than the well-known result stating that the essential commutant of the full Toeplitz algebra ${\mathfrak{T}}$ is ${\mathfrak{T}}$ (QC).