Additive mappings in algebras of unbounded operators preserving operators of rank one

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 $A\mathcal{D}\subset \mathcal{D},A^{\ast }\mathcal{D}\subset \mathcal{D}$. 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.