G-convergence of Dirac operators (Q416348)

Let \(\mathcal Y\) be a Hilbert space and let \(\lambda \geq 0\). Denote by \(\mathcal D_\lambda (\mathcal Y)\) the set of not necessarily densely defined operators \(A\) in \(\mathcal Y\) such that \(A\geq \lambda \) and \(A\) is self-adjoint in \(\overline{{\mathbf D}(A)}\). A sequence \((A_h)\) in \(\mathcal D_\lambda (\mathcal Y)\) is said to \(G\)-converge to \(A\) in \(\mathcal D_\lambda (\mathcal Y)\) if \(A_h^{-1}P_h\to A^{-1}P\) (in the strong or weak sense), where \(P\) is the orthogonal projection onto \({\mathbf D}(A)\). The authors show that \(G\)-convergence implies (weak) convergence of corresponding eigenvalues and eigenvectors. This result is applied to perturbations of Dirac operators, restricted to the spectral subspaces associated with the eigenvalues in the gap of the continuous spectrum.