Generalized Schur complements and P-complementable operators Original Research Article

Let A be a selfadjoint operator and P be an orthogonal projection both operating on a Hilbert space image. We say that A is P-complementable if A−μPgreater-or-equal, slanted0 holds for some image. In this case we define image. As a tool for computing IP(A) we introduce a natural generalization of the Schur complement or shorted operator of A to image, 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 image. We give some applications in the finite dimensional context.