Quasi-splitting subspaces and Foulis-Randall subspaces

For a pre-Hilbert space S, let F(S) denote the orthogonally closed subspaces, E_q(S) the quasi-splitting subspaces, E(S) the splitting subspaces, D(S) the Foulis-Randall subspaces, and R(S) the maximal Foulis-Randall subspaces, of S. It was an open problem whether the equalities D(S) = F(S) and E(S) = R(S) hold in general [Cattaneo, G. and Marino, G., “Spectral decomposition of pre-Hilbert spaces as regard to suitable classes of normal closed operators,” Boll. Unione Mat. Ital. 6 1-B, 451–466 (1982); Cattaneo, G., Franco, G., and Marino, G., “Ordering of families of subspaces of pre-Hilbert spaces and Dacey pre-Hilbert spaces,” Boll. Unione Mat. Ital. 71-B, 167–183 (1987); Dvurečenskij, A., Gleason's Theorem and Its Applications (Kluwer, Dordrecht, 1992), p. 243.]. We prove that the first equality is true and exhibit a pre-Hilbert space S for which the second equality fails. In addition, we characterize complete pre-Hilbert spaces as follows: S is a Hilbert space if, and only if, S has an orthonormal basis and E_q(S) admits a non-free charge.