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

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The spectrum of a selfadjoint quadratic operator pencil of the form λ²M-λG-A is investigated where M≥ 0, G≥ 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.