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

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 $n=-1,0,\dots$, ${\beta }_{-1}$ unitary, $\Vert {\beta }_j\Vert <1$ for $j\ge 0$ and ${\sum }_{j=0}^{\infty }\Vert {\beta }_j\Vert <\infty$; (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 $p\ge 1$. 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.