## Abstract

Suppose that H is a normal operator, the pencil L_{0} (λ) = I- λ^{n}H^{n} has a discrete and positive spectrum in the domain Ω(2θ, R) = (λ: larg λ < 2θ, λ > R). and S(λ) is an operator-valued function that is holomorphic in Ω(2θ. R.) and small in comparison to L_{0}(λ) (in a certain sense). A theorem is proved on comparison of the spectra of L(λ) = L_{0}(λ)- S(λ) and L_{0}(λ), i.e., on an estimate of the difference N(r) — N_{0}(r), where N(r) (N_{0}(r)) is the distribution function of the spectrum of L(λ) (L_{0}(λ)) in Ω(θ, p) (p ≥ R). This result implies generalizations of theorems of Keldysh on the asymptotic behavior of the spectrum of a polynomial operator pencil.

Original language | English |
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Pages (from-to) | 389-404 |

Number of pages | 16 |

Journal | Mathematics of the USSR - Sbornik |

Volume | 51 |

Issue number | 2 |

DOIs | |

State | Published - 28 Feb 1985 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics