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

Abstract

This paper is dedicated to Barry Simon on the occasion of his 60th birthday in appreciation for all that he has taught us. AbstractIn 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.Let d be a probability measure on the unit circle = {z ∈ C : |z| = 1} and let n = z n + · · ·, n 0, be the (monic) orthogonal polynomials (OPUCs) associated with d , m (z) n (z) d = 0, m = n, m, n 0 (see [20]). Let = ( n ) n∈Z + denote the vector of Verblunsky coefficients n = − n+1 (0), n 0. By Verblunsky's theorem (see [17]), the map V : d → is a bijection from the probability measures on onto × ∞ j =0 D, where D = {z ∈ C : |z| < 1} is the (open) unit disc in C. Following Cantero et al.[5], we may, given , construct a (pentadiagonal) unitary matrix operator U = U( ) in l 2 + = l 2 (Z + ) (the so-called CMV matrix) with the following property: e 0 = (1, 0, . . . ) T is a cyclic vector for U, i.e. U k e 0 −∞<k<∞ = l 2 + , and the associated spectral

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

“…This result is due to Deift-Ostensson [4], but [12] has a new proof. Earlier, [10] proved the weaker result when R 3 is replaced by R 2 .…”

“…The main theorems of [12] are: Theorem 1.3 [4]. If lim sup|α n | 1/n = R −1 < 1, then r(z) − S(z) is analytic in {z : 1 − δ < |z| < R 3 } for some δ > 0.…”

On extensions of a theorem of Baxter

“…An important fact, established independently in [2,13,8] (cf. Corollary 2 therein) is that for n ∈ N,…”

Orthogonal polynomials with exponentially decaying recursion coefficients

“…Simon [18] proved that r(z) -S(z) is analytic in {z I 1 -E < Izi < R2} when (3.1) holds, thereby generalizing [1]. The ultimate result of this genre was found independently by Deift-Ostensson [7] and Martinez-Finkelshtein et al [13]; an alternate proof was then found by Simon [20].…”