## Abstract

Let a selfadjoint operator-valued function L(λ) be given on the interval [a,b] such that L(a)≪0, L(b)≫0, L′(λ)≫0 (a≤λ≤b), and L″(λ) has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued function L(λ) can be reduced to the spectral theory of one operator Z, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L(λ)=M(λ)(λI-Z), where the operator-valued function M(λ) is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition of L^{-1}(λ) into the sume of its principal and regular parts.

Original language | English |
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Pages (from-to) | 716-746 |

Number of pages | 31 |

Journal | Integral Equations and Operator Theory |

Volume | 14 |

Issue number | 5 |

DOIs | |

State | Published - 1 Sep 1991 |

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory