  B. Simon proved the existence of the wave operators for the CMV matrices with
Szego class Verblunsky coefficients, and therefore the existence of the
scattering function. Generally, there is no hope to restore a CMV matrix when
we start from the scattering function, in particular, because it does not
contain any information about the (possible) singular measure. Our main point
of interest is the solution of the inverse scattering problem (the heart of the
Faddeev--Marchenko theory), that is, to give necessary and sufficient
conditions on a certain class of CMV matrices such that the restriction of this
correspondence (from a matrix to the scattering function) is one to one. In
this paper we show that the main questions on inverse scattering can be solved
with the help of three important classical results: Adamyan-Arov-Krein (AAK)
Theory, Helson-Szego Theorem and Strong Szego Limit Theorem. Each of these
theorem states the equivalence of certain conditions. Actually, to each theorem
we add one more equivalent condition related to the CMV inverse scattering
problem.
