## Abstract

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblun-sky parameters of the CMV matrices, corresponding spectral measures and scattering functions.

Original language | English |
---|---|

Pages (from-to) | 479-508 |

Number of pages | 30 |

Journal | Integral Equations and Operator Theory |

Volume | 69 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2011 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© Springer Basel AG 2011.

## Keywords

- Arov regularity
- Faddeev-Marchenko space
- Gelfand-Levitan-Marchenko equation
- Scattering function
- Schur algorithm
- Spectral measure
- Transformation operator
- γ-generating pair