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B. Simon proved the existence of the wave operators for the CMV matrices withSzego class Verblunsky coefficients, and therefore the existence of thescattering function. Generally, there is no hope to restore a CMV matrix whenwe start from the scattering function, in particular, because it does notcontain any information about the (possible) singular measure. Our main pointof interest is the solution of the inverse scattering problem (the heart of theFaddeev--Marchenko theory), that is, to give necessary and sufficientconditions on a certain class of CMV matrices such that the restriction of thiscorrespondence (from a matrix to the scattering function) is one to one. Inthis paper we show that the main questions on inverse scattering can be solvedwith the help of three important classical results: Adamyan-Arov-Krein (AAK)Theory, Helson-Szego Theorem and Strong Szego Limit Theorem. Each of thesetheorem states the equivalence of certain conditions. Actually, to each theoremwe add one more equivalent condition related to the CMV inverse scatteringproblem.

Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem