Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W ∈ L(H) a positive operator such that W

Given a bounded selfadjoint operator W on a Krein space H and a closed subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators.

Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W ∈ L(H) a positive operator such that W 1/2 is in the p-Schatten class, for some 1 ≤ p < ∞. Given A, B ∈ L(H) with closed range and C ∈ L(H), we study the following weighted approximation problem: analize the existence of min X∈L(H) AXB − C p,W , (0.1) where X p,W = W 1/2 X p. We also study the related operator approximation problem: analize the existence of min X∈L(H) (AXB − C) * W (AXB − C), (0.2) where the order is the one induced in L(H) by the cone of positive operators. In this paper we prove that the existence of the minimum of (0.2) is equivalent to the existence of a solution of the normal equation A * W (AXB − C) = 0. We also give sufficient conditions for the existence of the minimum of (0.1) and we characterize the operators where the minimum is attained.

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