# 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

## Abstract

In this survey we establish bijective correspondences between the following classes of objects: (1) $\beta_{-1}$ and $\{ \beta_n \}_{n=0}^{\infty}$, with $\beta_n \in \mathbb C^{p \times p}$ for $n=-1,0,\ldots$, $\beta_{-1}$ unitary, $\| \beta_j \| < 1$ for $j \geq 0$ and $\sum_{j=0}^{\infty} \| \beta_j \| < \infty$; (2) A unitary matrix $\beta_{-1} \in \mathbb C^{p \times p}$ and a spectral density $\Delta$ belonging to the Wiener algebra $\mathcal W^{p \times p}$ with $\Delta(\zeta) \succ 0$ for all $\zeta$ on the unit circle $\mathbb T$; (3) CMV matrices based on a unitary matrix $\beta_{-1} \in \mathbb C^{p \times p}$ and a spectral density $\Delta$ that meets the constraints in (2); (4) scattering matrices that belong to the Wiener algebra $\mathcal W^{p \times p}$; (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 $\mathcal W^{p \times p}$ (i.e., between class (1) and class (2)) is known as Baxter's theorem and was established by Baxter when $p=1$ and Geronimo when $p \geq 1$. The connections between CMV matrices, the solutions of a related Nehari problem and an inverse scattering problem seem to be new when $p > 1$. 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