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

### T.A. Mel'nyk

Kyiv University, Ukraine

## Abstract

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