Oblique projections and Schur complements

LetH be a Hilbert space, L(H) the algebra of all bounded linear operators onH 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 = QA} for different choices of A, mainly under the hypothesis that A ≥ 0. There is a closed relationship between the A-selfadjoint projections onto S and the shorted operator (also called Schur complement) of A to S⊥. Using this relation we find several conditions which are equivalent to the fact that P(A,S) 6= ∅, in particular in the case of A ≥ 0 with A injective or with R(A) closed. If A is itself a projection, we relate the set P(A,S) with the existence of a projection with fixed kernel and range and we determine its norm.