Hausdorff convergence and asymptotic estimates of the spectrum of a perturbed operator
Summary: A family of self-adjoint compact operators \(A_\varepsilon\) \((\varepsilon>0)\) acting in Hilbert spaces \({\mathcal H}_\varepsilon\) is considered. The asymptotic behaviour as \(\varepsilon\to 0\) of eigenvalues and eigenvectors of the operators \(A_\varepsilon\) 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 \(A_\varepsilon\) and points of the spectrum \(\sigma(A_0)\) (both of the discrete spectrum and the essential one) are obtained. Asymptotic estimates for eigenvectors of \(A_\varepsilon\) are also proved.
self-adjoint compact operators in Hilbert spaces
scientific article; zbMATH DE number 1747398
scientific article