Schur Complements in Krein Spaces

Abstract.

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space ${\mathcal{H}}$ and a suitable closed subspace ${\mathcal{S}}$ of ${\mathcal{H}}$, the Schur complement $A_{/[\mathcal{S}]}$ of A to ${\mathcal{S}}$ is defined. The basic properties of $A_{/[\mathcal{S}]}$ are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space.