JournalszaaVol. 20, No. 4pp. 941–957

Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator

  • T.A. Mel'nyk

    Kyiv University, Ukraine
Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator cover
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Abstract

A family of self-adjoint compact operators Aϵ(ϵ>0)A_{\epsilon} (\epsilon > 0) acting in Hilbert spaces Hϵ\mathcal H_{\epsilon} is considered. The asymptotic behaviour as ϵ0\epsilon \to 0 of eigenvalues and eigenvectors of the operators AϵA_{\epsilon} is studied; the limiting operator A0:H0H0A_0 : \mathcal H_0 \mapsto \mathcal H_0 is non-compact. Asymptotic estimates of the differences between eigenvalues of AϵA_{\epsilon} and points of the spectrum σ(A0)\sigma (A_0) (both of the discrete spectrum and the essential one) are obtained. Asymptotic estimates for eigenvectors of AϵA_{\epsilon} are also proved.

Cite this article

T.A. Mel'nyk, Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator. Z. Anal. Anwend. 20 (2001), no. 4, pp. 941–957

DOI 10.4171/ZAA/1053