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

Abstract

A family of self-adjoint compact operators $A_{ϵ}(ϵ>0)$ acting in Hilbert spaces ${\mathcal{H}}_{ϵ}$ is considered. The asymptotic behaviour as $ϵ\to 0$ of eigenvalues and eigenvectors of the operators $A_{ϵ}$ 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 $A_{ϵ}$ and points of the spectrum $\sigma (A_0)$ (both of the discrete spectrum and the essential one) are obtained. Asymptotic estimates for eigenvectors of $A_{ϵ}$ are also proved.