Generalized Schur Complements and P-complementable Operators

P. MASSEY Y D. STOJANOFF

LINEAR ALGEBRA AND ITS APPLICATIONS

Birkhauser

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 $mathcal{H}$. We say that $A$ is $P-complementable$ if   $A - \mu P \geq 0$ holds for some $\mu \in \mathbb{R}$. In this case we define $I_{P}(A)= \mathrm{\max}{\mu \in \mathbb{R}: A-\mu P \geq 0 }$. As a tool for computing $I_{P}(A)$ we introduce a natural generalization of the Schur complement or shorted operator of $A$ to $\mathcal{S}=R(P)$, denoted by $\Sigma(A,P)$. 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 $\Sigma(A,P)$ for $P$-complementable $A$, under the hypothesis of compatibility of the pair $(A, \mathcal{S})$}. We give some applications in the finite dimensional context.