A Riemann–Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials

Abstract

In this paper, the authors show how to use Riemann–Hilbert techniques to prove various results, some old, some new, in the theory of Toeplitz operators and orthogonal polynomials on the unit circle (OPUCs). There are four main results: the first concerns the approximation of the inverse of a Toeplitz operator by the inverses of its finite truncations. The second concerns a new proof of the ‘hard’ part of Baxter's theorem, and the third concerns the Born approximation for a scattering problem on the lattice Z + . The fourth and final result concerns a basic proposition of Golinskii–Ibragimov arising in their analysis of the Strong Szegö Limit Theorem.