Summary: We consider a selfadjoint operator function \(L\) of the form \(L(\lambda):= \lambda- A\pm B^*(C- \lambda)^{- 1} B\) under the assumption that the spectrum of \(L\) splits into two parts. In case of the sign \(+\) with the pencil \(L\) there is associated a selfadjoint operator \(\widetilde A\) in some Hilbert space \(\widetilde{\mathcal H}\supset {\mathcal H}\), in case of the sign \(-\) there is associated with \(L\) a selfadjoint \(\widetilde B\) in a Kreĭn space \(\widetilde{\mathcal K}\supset {\mathcal H}\).
Spectral properties of these associated operators are crucial for the study of the spectral properties of \(L\). Sufficient conditions for the fact that the eigenvectors corresponding to certain parts of the spectrum of \(L\) form a Riesz basis in \(\mathcal H\) are given.