Generalized Schur complements and P-complementable operators

Abstract

Let A be a selfadjoint operator and P be an orthogonal projection both operating on a Hilbert space H. We say that A is P-complementable if A−µP≥ 0 holds for some µ ∈ R. In this case we define IP (A) = max{µ ∈ R : A − µP ≥0}. As a tool for computing IP(A) we introduce a natural generalization of the Schur complement or shorted operator of A to f A to S = R(P ), denoted by Σ(A, P ). We give expressions and a characterization for IP(A) that generalize some known results for particular choices of P. We also study some aspects of the shorted operator Σ(A,P) for P-complementable A, under the hypothesis of compatibility of the pair. We give some applications in the finite dimensional context.