Indefinite least-squares problems and pseudo-regularity

Given two Krein spaces HH and KK, a (bounded) closed-range operator C:H→KC:H→K and a vector y∈Ky∈K, the indefinite least-squares problem consists in finding those vectors u∈Hu∈H such that[Cu−y,Cu−y]=minx∈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 KK. Along this article the range of C is only supposed to be a J  -nonnegative pseudo-regular subspace of KK. 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.