Projections in operator ranges

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