On the limit behavior of the spectrum of a sequence of operators given in various Hilbert spaces. (Russian) Zbl 0684.47002

Iosif’yan, G. A.; Oleĭnik, O. A.; Shamaev, A. S.

Usp. Mat. Nauk 44, No. 3(267), 157-158 (1989).

For separable Hilbert spaces \({\mathcal H}_{\epsilon}\), \({\mathcal H}_ 0\) (\(\epsilon\) is a small parameter), close in a certain sense (defined with the help of operators \(R_{\epsilon}:{\mathcal H}_ 0\to {\mathcal H}_{\epsilon})\) selfadjoint compact and positive operators \({\mathcal A}_{\epsilon}\in {\mathcal L}({\mathcal H}_{\epsilon})\) and \({\mathcal A}_ 0\in {\mathcal L}({\mathcal H}_ 0)\) are considered. If the operators and spaces satisfy the precisely formulated C1-C4 conditions, estimations for convergence of the eigenvalues \(\mu^ k_{\epsilon}-\mu^ k_ 0\) of operators are given. Similar assertion is proved for the eigenvectors of the operators.

Reviewer: R.Duducava

MSC:

47A10

Spectrum, resolvent

Keywords:

eigenvalues; eigenvectors; small parameter; selfadjoint compact and positive operators