On the theory of \(J\)-selfadjoint perturbations of self-adjoint operators
On the direct sum \({\mathcal H} \equiv H\oplus H\) of a Hilbert space \(H\), the author considers an indefinite inner product induced by \(J = \left(\begin{smallmatrix} I & 0 \\ 0 & -I \end{smallmatrix}\right) \). Let \(A_i\) \((i = 1,2)\) be completely continuous self-adjoint operators on \(H\) and \({\mathcal K}\) a finite rank \(J\)-selfadjoint operator on \({\mathcal H}\). For a perturbation of the form \[ {\mathcal A}(\varepsilon) = {\mathcal A} + \varepsilon{\mathcal K}, \quad {\mathcal A} = \begin{pmatrix} A_1 & 0 \\ 0 & A_2\end{pmatrix} \] the author considers the problem whether for small \(\varepsilon > 0\) the operator \({\mathcal A}(\varepsilon)\) has at most finite numbers of non real eigenvalues provided \(A_1\) and \(A_2\) have no common eigenvalues. By a simple example he shows that this is not the case even when \({\mathcal K}\) is of rank one .
scientific article; zbMATH DE number 1002982