Semiclosed projections and applications

Abstract

We characterize the semiclosed projections and apply them to compute the Schur complement of a selfadjoint operator with respect to a closed subspace. These projections occur naturally when dealing with weak complementability.

Idempotent linear relations

“…Further investigations on closed densely defined projections were carried out by Andô [1]. We extended this work to semiclosed projections in [2].…”

“…So a semi-projection is a projection if and only if (2) holds. Dropping not just (2), but both (1) and ( 2), the result is an idempotent relation; that is, a linear relation E such that E 2 = E.…”

Matrix representations of multivalued projections and least squares problems

“…Another example of a semiclosed multivalued projection appears when solving operator least squares problems with a selfadjoint or positive (semi-definite) weight in a Hilbert space [6]. This is explained in Example 4. It is easy to check that a multivalued projection is semiclosed if and only if its range and kernel are both operator ranges, where the special case for operators can be found in [26] and [9].…”