Hermitian pencils and their representation in Krein spaces

Pencils of the form A(λ) = λE−A are studied, where A and E are bounded linear operators on a Hilbert space. Of interest are the spectral properties of A(λ). This is done via a corresponding linear relation in a Krein space, which is given in range representation using the two operators A and E. Under some assumptions on E and A, the linear relation in range representation is nonnegative or has finitely many negative squares. Then one uses spectral properties of linear relations and deduces spectral properties of the operator pencil A(λ) = λE−A.