Indefinite least-squares problems and pseudo-regularity

Abstract

Given two Krein spaces H and K, a (bounded) closed-range operator C:H→K and a vector y∈K, the indefinite least-squares problem consists in finding those vectors u∈H such that [Cu - y, Cu - y] = min_x∈H[Cx - y, Cx - y]. The indefinite least-squares problem has been thoroughly studied before under the assumption that the range of C is a uniformly J-positive subspace of K. Along this article the range of C is only supposed to be a J-nonnegative pseudo-regular subspace of K. This work is devoted to present a description for the set of solutions of this abstract problem in terms of the family of J-normal projections onto the range of C.