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

Let H be a complex Hilbert space, and let S-a (H) be the space of all self-adjoint operators on H. Let Gamma denote the subset of S, (H) consisting of all rank one operators. A map L from Sa (H) to itself is called a rank one preserver on S-a (H) if L (Gamma) subset of Gamma, and is said to be R-linear if L (A + B) = L (A) + L (B) and L (kA) = kL (A) for any k is an element of R and A, B is an element of S-a (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 S,(H). Moreover, we also determine the general form of all R-linear and weakly continuous rank preservers (respectively, rank one idempotence preservers) on S-a (H). (c) 2005 Elsevier Inc. All rights reserved.