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 script H sign. We say that A is P-complementable if A-μP≥0 holds for some μ∈R. In this case we define I P(A)=max{μ∈R:A-μP≥0}. As a tool for computing I P(A) we introduce a natural generalization of the Schur complement or shorted operator of A to script S sign=R(P), denoted by Σ(A,P). We give expressions and a characterization for I P(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 (A,script S sign). We give some applications in the finite dimensional context.