Generalized Orthogonal Projections and Shorted Operators

Let H be a Hilbert space, L(H) the algebra of all bounded linear operators on H and 〈 , 〉A : H×H → C the bounded sesquilinear form induced by a selfadjoint A ∈ L(H), 〈ξ, η〉A = 〈Aξ, η〉, ξ, η ∈ H. Given T ∈ L(H), T is A-selfadjoint if AT = T ∗A. If S ⊆ H is a closed subspace, we study the set of A-selfadjoint projections onto S, P(A,S) = {Q ∈ L(H) : Q = Q , R(Q) = S , AQ = Q∗A} for different choices of A, mainly under the hypothesis that A ≥ 0. In this paper we study the close relationship between the existence and properties of A-selfadjoint projections onto S and the shorted operator (also called Schur complement) A/S of A to S and the S-compression AS = A −A/S .