The tame symbol and determinants of Toeplitz operators

TL;DR

This paper derives an explicit formula for the determinant of a product of Toeplitz operators and their inverses using tame symbols, linking operator theory with complex analysis.

Contribution

It introduces a novel explicit formula connecting Toeplitz operator determinants with tame symbols of their symbols on the unit disk.

Findings

Explicit determinant formula involving tame symbols

Connection between Toeplitz operators and algebraic symbols

Extension of classical results to meromorphic symbols

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.