A matrix formula for Schur complements of nonnegative selfadjoint linear relations

Abstract

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