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