## Abstract

We study finite-dimensional perturbations A + γB of a self-adjoint operator A acting in a Hilbert space Heng hooktop sign. We obtain asymptotic estimates of eigenvalues of the operator A+γB in a gap of the spectrum of the operator A as γ → 0, and asymptotic estimates of their number in that gap. The results are formulated in terms of new notions of characteristic branches of A with respect to a finite-dimensional subspace of Heng hooktop sign on a gap of the spectrum σ(A) and asymptotic multiplicities of endpoints of that gap with respect to this subspace. It turns out that if A has simple spectrum then under some mild conditions these asymptotic multiplicities are not bigger than one. We apply our results to the operator (Af)(t) = tf(t) on L^{2}([0, 1],p_{c}), where p_{c} is.the Cantor measure, and obtain the precise description of the asymptotic behavior of the eigenvalues of A + γB in the gaps of σ(A) = script C sign(= the Cantor set).

Original language | English |
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Pages (from-to) | 127-164 |

Number of pages | 38 |

Journal | Integral Equations and Operator Theory |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - 1999 |

### Bibliographical note

Funding Information:Supported by a grant from the German-Israeli Foundation (GIF)

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory