Hermitian Pencils and their Representation in Krein Spaces

Abstract

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