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 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 S=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, S) . We give some applications in the finite dimensional context.