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: find u∈ℋ such that 〈B(Cu−y),Cu−y〉=min x∈ℋ〈B(Cx−y),Cx−y, where B:K→K is a bounded selfadjoint operator. 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.