Projections in operator ranges

Abstract

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