On spectra of quadratic operator pencils with rank one gyroscopic linear part

The spectrum of a selfadjoint quadratic operator pencil of the form ${\lambda }^{2}M-\lambda G-A$ is investigated where $M\ge 0$, $G\ge 0$ are bounded operators and $A$ is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator $G$ the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator $G$. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators $A$ are given.