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

AbstractLet 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)