2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71362Let $ \mathcal D$ be a dense linear manifold in a Hilbert space $\mathcal H$ and let $L^+(\mathcal D)$ be the *-algebra of all linear operators $A$ such that $A \mathcal D \subset \mathcal D, A^* \mathcal D \subset \mathcal D$. Denote by $F(\mathcal D)$ the *-ideal of $L^+(\mathcal D)$ consisting of all finite-rank operators. A characterization of the structure of additive mappings of $F(\mathcal D)$ preserving operators of rank one or projections of rank one is given. The corresponding results for algebras of bounded operators on Banach spaces were given by P. uSemrl.7 pages The first version of the paper is replaced by this one, because in the first version the enumeration of the Theorems, Lemmas etc. were incorrectOperator Algebras47L60text