LAUSR

Abstract We prove the following theorem. Let T be a dense embedding of a Hilbert space H into a Banach space Y. Assume that there exists a subspace X of H whose density character satisfies dens X = dens X ⊥ , such that T | X is an isomorphism and T ( X ) is complemented to the closure T ( X ⊥ ) ‾ . Then there is a subspace E of H for which both restrictions T | E and T | E ⊥ are isomorphisms. This result is applied to concrete spaces. Dual and almost proportional versions are also considered. A weaker version gives a complete description of Banach spaces having quasicomplemented subspaces which are isomorphic to Hilbert spaces.