Schur complements of selfadjoint Krein space operators

TL;DR

This paper investigates the properties of Schur complements for bounded selfadjoint operators on Krein spaces, providing a variational characterization and identifying conditions for their existence.

Contribution

It introduces a variational approach to Schur complements in Krein spaces and characterizes operators that admit such complements.

Findings

Variational characterization of Schur complements

Identification of S-weakly complementable operators

Conditions for the existence of Schur complements in Krein spaces

Abstract

Given a bounded selfadjoint operator W on a Krein space H and a closed subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators.