Oblique projections and Schur complements

Abstract

Let H be a Hilbert space, L(H) the algebra of all bounded linear operators on H and ⟨,⟩_A : H x H → C the bounded sesquilinear form induced by a selfadjoint A ∈ L(H), ⟨ξ, n⟩_A =⟨Aξ, n⟩, ξ , n ∈ 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^2 = Q, R(Q) = S , AQ = Q^*A 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 areequivalent to the fact that P(A, S), 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.