Upper semicontinuity of eigenvalues of selfadjoint operators defined on moving domains (Q1070448)

The author considers a sequence of positive self-adjoint operators \(A_{\epsilon}\) defined in a sequence of Hilbert spaces \(V_{\epsilon}\subset X\) which depend on a parameter \(\epsilon\). Here \(\epsilon\) denotes a sequence decreasing to zero while X is a real, separable and infinitely dimensional Hilbert space, such that the injection \(V_{\epsilon}\to X\) is compact. The range of \(A_{\epsilon}\) is supposed to be in the closure of \(V_{\epsilon}\) in X. Then he shows that all the eigenvalues of \(A_{\epsilon}\), say \(\mu_{\epsilon}^{(k)}\), \(k=1,2,...\), are upper semicontinuous under certain conditions, i.e.: \(\limsup_{\epsilon \to 0}\mu_{\epsilon}^{(k)}\leq \mu_ 0^{(k)}\), \(k=1,2,... \).