Linear Maps Preserving Projections of Jordan Products on the Space of Self-Adjoint Operators

Abstract:

      Let ${\mathcal {B}}_ {s}(\mathcal {H})$ be the real linear space of all self-adjoint operators on a complex Hilbert space $\mathcal {H}$ with $\dim {\mathcal {H}}\geq 2.$ It is proved that a linear surjective map $\varphi$ on ${\mathcal {B}}_{s}(\mathcal {H})$ preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator $U$ on $\mathcal {H}$ such that $\varphi(X)=\lambda U^*XU, \forall X \in {\mathcal {B}}_{s}(\mathcal {H})$ for some constant $\lambda$ with $\lambda\in\{1,-1\}.$