Projections in operator ranges

Abstract

If H is a Hilbert space, A is a positive bounded linear operator on H and S is a closed subspace of H, the relative position between S and the A-1 (S⊥) establishes a notion of compatibility. We show that the compatibility of (A,S) is equivalent to the existence of a convenient orthogonal projection in the operator range R(A 1/2), with the canonical Hilbertian structure.