CMV matrices with asymptotically constant coefficients. Szeg\"o over Blaschke class, Scattering Theory

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We demonstrate that an orthonormal system in a certain "weighted'' Hilbert space, which we call the Fadeev-Marchenko (FM) space, behaves asymptotically as the system in the standard (free) case. The duality between the two types of Hardy subspaces in it plays the key role in the proof of all asymptotics involved. We show that the traditional (Faddeev-Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: 1) Szeg\"o-Blaschke class: the class of twosided CMV matrices acting in $l^2$, whose spectral density satisfies the Szeg\"o condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate the scattering data and related to them the FM space. The CMV matrix corresponds to the multiplication operator in this space, and the orthonormal basis in it (corresponding to the standard basis in $l^2$) behaves asymptotically as the basis associated with the free system. 2) $A_2$-Carleson class: from the point of view of the scattering problem, the most natural class of CMV matrices is that one in which a) the scattering data determine the matrix uniquely and b) the associated Gelfand- Levitan- Marchenko transformation operators are bounded. Necessary and sufficient conditions for this class can be given in terms of an $A_2$ kind condition for the density of the absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar close to the optimal conditions are given directly in terms of the scattering data.