2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161982Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case.Functional Analysis46B03; 46C15; 47B99text