A family of self-adjoint compact operators acting in Hilbert spaces is considered. The asymptotic behaviour as of eigenvalues and eigenvectors of the operators is studied; the limiting operator is non-compact. Asymptotic estimates of the differences between eigenvalues of and points of the spectrum (both of the discrete spectrum and the essential one) are obtained. Asymptotic estimates for eigenvectors of are also proved.
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T.A. Mel'nyk, Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator. Z. Anal. Anwend. 20 (2001), no. 4, pp. 941–957DOI 10.4171/ZAA/1053