In this paper we study A-projections, i.e. operators of a Hilbert space H which act as projections when a seminorm is considered in H. A-projections were introduced by Mitra and Rao [21] for finite dimensional spaces. We relate this concept to the theory of compatibility between positive operators and closed subspaces of H. We also study the relationship between weighted least squares problems and compatibility.
We study the minus order on the algebra of bounded linear operators on a Hilbert space. By giving a characterization in terms of range additivity, we show that the intrinsic nature of the minus order is algebraic. Applications to generalized inverses of the sum of two operators, to systems of operator equations and to optimization problems are also presented.
Different equivalence relations are defined in the set L(H) s of selfadjoint operators of a Hilbert space H in order to extend a very well known relation in the cone of positive operators. As in the positive case, for a ∈ L(H) s the equivalence class C a admits a differential structure, which is compatible with a complete metric defined on C a. This metric coincides with the Thompson metric when a is positive.
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