Generalized Schur complements and $P$-complementable operators

MASSEY, PEDRO; STOJANOFF, DEMETRIO

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER

Año: 2004 vol. 393 p. 299 - 318

0024-3795

Let A be a selfadjoint operator and P be an orthogonal projection both operating on a Hilbert space . We say that A is P-complementable if A−$mu$P0 holds for some . In this case we define                                                    As a tool for computing IP(A) we introduce a natural generalization of the Schur complement or shorted operator of A to , denoted by $Sigma$(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  $Sigma$(A,P) for P-complementable A, under the hypothesis of compatibility of the pair . We give some applications in the finite dimensional context.