自伴算子空间上保持Jordan积投影的线性映射

Let Bs(H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dim H ≥ 2.It is proved that a linear surjective map on Bs (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 H such that (X)=λU XU,X∈Bs(H) for some constant λ with λ∈{1,1}.

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

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 $\lambda\in\{1,-1\}.$