A matrix formula for Schur complements of nonnegative selfadjoint linear relations

TL;DR

This paper derives explicit matrix formulas for Schur complements and compressions of nonnegative selfadjoint linear relations in Hilbert spaces, under invariance conditions, extending classical matrix analysis to operator relations.

Contribution

It provides a new matrix representation for such relations and explicit formulas for their Schur complements and compressions.

Findings

Explicit matrix formulas for Schur complements of nonnegative selfadjoint relations.

Representation of relations with respect to invariant subspaces.

Extension of classical matrix results to operator relations.

Abstract

If a nonnegative selfadjoint linear relation $A$ in a Hilbert space and a closed subspace $\mathcal{S}$ are assumed to satisfy that the domain of $A$ is invariant under the orthogonal projector onto $\mathcal{S},$ then $A$ admits a particular matrix representation with respect to the decomposition $\mathcal{S} \oplus \mathcal{S}^{\perp}$. This matrix representation of $A$ is used to give explicit formulae for the Schur complement of $A$ on $\mathcal{S}$ as well as the $\mathcal{S}-$compression of $A$.