Schur complements of selfadjoint Krein space operators

Abstract

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.

Weighted operator least squares problems and the J‐trace in Krein spaces

“…In this section we include several results on the Schur complement in Krein spaces that will be useful along the paper. For the proofs the reader is referred to [9].…”

“…In [9], the notions of ‐complementability, ‐weak complementability and the Schur complement were extended to the Krein space setting. In what follows is a Krein space.…”

“…In this case, if the matrix representation of induced by is the ‐weak complementability of W is equivalent to . The ‐weak complementability of W does not depend on the signature operator, see [9, Thm. 4.4].…”

“…There the fundamental tool for solving the least squares problems was given by the indefinite inverse. In this work the Schur complement, as defined and studied in [9], plays this role.…”

“…The paper has four additional sections. Section 2 fixes notation and recalls the basics of Krein spaces, Section 3 gives a brief account of the fundamental results on the Schur complement from [9]. In Section 4 we turn to weighted least squares problems.…”