Hermitian Pencils and their Representation in Krein Spaces

Pencils of the form $\mathcal{A}({\lambda}) = {\lambda}E-A$ are studied, where $A$ and $E$ are bounded linear operators on a Hilbert space.

Of interest are the spectral properties of $\mathcal{A}({\lambda})$.

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 $\mathcal{A}({\lambda}) = {\lambda}E-A$.