## Abstract

In spectral theory, j-monotonic families of 2 × 2 matrix functions appear as transfer matrices of many one-dimensional operators. We present a general theory of such families, in the perspective of canonical systems in Arov gauge. This system resembles a continuum version of the Schur algorithm, and allows to restore an arbitrary Schur function along the flow of associated boundary values at infinity. In addition to results in Arov gauge, this provides a gauge-independent perspective on the Krein–de Branges formula and the reflectionless property of right limits on the absolutely continuous spectrum. This work has applications to inverse spectral problems which have better behavior with respect to a normalization at an internal point of the resolvent domain.

Original language | English |
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Article number | 4 |

Journal | Integral Equations and Operator Theory |

Volume | 94 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2022 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

### Funding

The work of R.B. in Sections 6 and 7 is supported by grant RScF 19-11-00058 of the Russian Science Foundation. In the rest of the paper, M.L. was supported in part by NSF grant DMS–1700179 and P.Y. was supported by the Austrian Science Fund FWF, project no: P32885-N.

Funders | Funder number |
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National Science Foundation | DMS–1700179 |

Austrian Science Fund | P32885-N |

Russian Science Foundation |

## Keywords

- Almost periodic measures
- Arov gauge
- Breimesser–Pearson theorem
- Canonical Hamiltonian systems
- Krein–de Branges formula
- Reflectionless
- Ricatti equation