Inversion of Toeplitz operators, Levinson equations, and Gohberg-Krein factorization—A simple and unified approach for the rational case

Abstract

The problem of the inversion of the Toeplitz operator T Φ , associated with the operator-valued function Φ defined on the unit circle, is known to involve the associated Levinson system of equations and the Gohberg-Krein factorization of Φ. A simplified and self-contained approach, making clear the connections between these three problems, is presented in the case where Φ is matrix-valued and rational. The key idea consists in looking at the Levinson system of equations associated with Φ −1( z −1), rather than that associated with Φ( z). As a consequence, a new invertibility criterion for Toeplitz operators with rational matrix-valued symbols is derived.