A generalized Schur complement for non-negative operators on linear space

TL;DR

This paper introduces a generalized Schur complement for non-negative linear operators on linear spaces, extending the concept from matrices and Hilbert space operators to a broader setting.

Contribution

It defines and explores properties of a new generalized Schur complement applicable to non-negative operators on linear spaces, broadening existing mathematical frameworks.

Findings

Established a new definition for the generalized Schur complement in this context.

Derived fundamental properties and theoretical insights of the generalized Schur complement.

Extended the applicability of Schur complement concepts beyond Hilbert spaces.

Abstract

Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space we define a generalized Schur complement for a non-negative linear operator mapping a linear space into its dual and derive some of its properties.