  Let $ \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. \u{S}emrl.
