The tame symbol and determinants of Toeplitz operators

Abstract

Suppose that $\phi$ and $\psi$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_\phi$ and $T_\psi$ denote the Toeplitz operators with symbols $\phi$ and $\psi$ respectively. We give an explicit formula for the determinant of $T_\phi T_\psi T_\phi^{-1} T_\psi^{-1}$ in terms of the products of the tame symbols of $\phi$ and $\psi$ on the open unit disk.