Rank one preserving R-linear maps on spaces of self-adjoint operators on complex Hilbert spaces

Let H be a complex Hilbert space, and let Sa(H) be the space of all self-adjoint operators on H. Let Γ denote the subset of Sa(H) consisting of all rank one operators. A map L from Sa(H) to itself is called a rank one preserver on Sa(H) if L(Γ) ⊆ Γ, and is said to be R-linear if L(A + B) = L(A) + L(B) and L(kA) = kL(A) for any k ∈ R and A, B ∈ Sa(H), where R is the field of real numbers. We give a complete classification of all R-linear and weakly continuous rank one preservers on Sa(H). Moreover, we also determine the general form of all R-linear and weakly continuous rank preservers (respectively, rank one idempotence preservers) on Sa(H).