JournalsemssVol. 3 , No. 1pp. 1–105

CMV matrices, a matrix version of Baxter's theorem, scattering and de Branges spaces

  • Harry Dym

    The Weizmann Institute of Science, Rehovot, Israel
  • David P. Kimsey

    Ben-Gurion University of the Negev, Beer-Sheva, Israel
CMV matrices, a matrix version of Baxter's theorem, scattering and de Branges spaces cover
Download PDF

A subscription is required to access this article.

Abstract

In this survey we establish bijective correspondences between the following classes of objects: (1) β1\beta_{-1} and {βn}n=0\{ \beta_n \}_{n=0}^{\infty}, with βnCp×p\beta_n \in \mathbb C^{p \times p} for n=1,0,n=-1,0,\ldots, β1\beta_{-1} unitary, βj<1\| \beta_j \| < 1 for j0j \geq 0 and j=0βj<\sum_{j=0}^{\infty} \| \beta_j \| < \infty; (2) A unitary matrix β1Cp×p\beta_{-1} \in \mathbb C^{p \times p} and a spectral density Δ\Delta belonging to the Wiener algebra Wp×p\mathcal W^{p \times p} with Δ(ζ)0\Delta(\zeta) \succ 0 for all ζ\zeta on the unit circle T\mathbb T; (3) CMV matrices based on a unitary matrix β1Cp×p\beta_{-1} \in \mathbb C^{p \times p} and a spectral density Δ\Delta that meets the constraints in (2); (4) scattering matrices that belong to the Wiener algebra Wp×p\mathcal W^{p \times p}; (5) a class of solutions of an associated matricial Nehari problem.

The bijective correspondence between summable sequences of contractions and positive spectral densities in the Wiener algebra Wp×p\mathcal W^{p \times p} (i.e., between class (1) and class (2)) is known as Baxter's theorem and was established by Baxter when p=1p=1 and Geronimo when p1p \geq 1. The connections between CMV matrices, the solutions of a related Nehari problem and an inverse scattering problem seem to be new when p>1p > 1. There is partial overlap of the connection between the considered Nehari problem and a discrete analogue of an inverse scattering problem considered by Krein and Melik-Adamjan. de Branges spaces of vector-valued polynomials are used to ease a number of computations.

Cite this article

Harry Dym, David P. Kimsey, CMV matrices, a matrix version of Baxter's theorem, scattering and de Branges spaces. EMS Surv. Math. Sci. 3 (2016), no. 1 pp. 1–105

DOI 10.4171/EMSS/14