2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76347If $\H$ 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.Functional Analysis46C07, 47A62, 46C05text