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Zeitschrift für Analysis und ihre Anwendungen


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Volume 20, Issue 4, 2001, pp. 941–957
DOI: 10.4171/ZAA/1053

Published online: 2001-12-31

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

T.A. Mel'nyk[1]

(1) Kyiv University, Ukraine

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.

Keywords: Spectrum, asymptotic estimates, perturbed operators

Mel'nyk T.A.: Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator. Z. Anal. Anwend. 20 (2001), 941-957. doi: 10.4171/ZAA/1053