Maps preserving numerical radius or cross norms of products of self-adjoint operators

Abstract

Let H be a complex Hilbert space with dim H ≥ 3, Bs(H) the (real) Jordan algebra of all self-adjoint operators on H. Every surjective map Φ: Bs(H) → Bs(H) preserving numerical radius of operator products (respectively, Jordan triple products) is characterized. A characterization of surjective maps on Bs(H) preserving a cross operator norm of operator products (resp. Jordan triple products of operators) is also given.