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

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

2005-04-13

arxiv.org/abs/math/0504284v1

Mathematics

Functional Analysis

Scientific paper

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 (OPUC's). 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 $\mathbb{Z}_+$. The fourth and final result concerns a basic proposition of Golinskii-Ibragimov arising in their analysis of the Strong Szeg\"{o} Limit Theorem.

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