A Geometrical Approach to Indefinite Least Squares Problems

Abstract

Given Hilbert spaces H and K, a (bounded) closed range operator C : H → K and a vector y ∈ K, consider the following indefinite least squares problem:This work is devoted to give necessary and sufficient conditions for the existence of solutions of this abstract problem. Although the indefinite least squares problem has been thoroughly studied in finite dimensional spaces, the geometrical approach presented in this manuscript is quite different from the analytical techniques used before. As an application we provide some new sufficient conditions for the existence of solutions of an H ∞ estimation problem.

Weighted projections into closed subspaces

“…Finally, by Proposition 4.9 and Remark 4.18, it holds that R(A 2 G) ⊆ N(A 1 B)) turns out to be compatible by [18,Proposition 3.9]. The converse follows by the previous result.…”

“…Again, the situation is completely described by using certain compatibility conditions. Analogous problems have been considered in [7] and [18]. It should also be mentioned that L. Eldén [16] was the first to study this problem in finite dimensions.…”

Indefinite least-squares problems and pseudo-regularity

“…In particular, Cx = y admits an ILSS for every y ∈ K if and only if R(C) is a uniformly J-positive subspace of K, see also [12,Proposition 3.2].…”

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

“…[19] has brought into play indefinite weighted least squares problems. Some of those problems were studied in their “pointwise” form, for linear operators on infinite‐dimensional spaces in [15] and, for matrices with complex entries in [18, 20]. Roughly speaking, if one is given an infinite or finite‐dimensional linear space , a weight W , bounded linear operators or matrices , and a vector , then the problem is to find an “extremal” vector for the quadratic form with a Krein space inner product on .…”