自伴算子空间上保持Jordan积投影的线性映射

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

中文关键词:  自伴算子  Jordan积  非零投影  线性映射.

英文关键词:self-adjoint operator  Jordan product  projection  linear map.

中文摘要:

      设$\mathcal {H}$是复的Hilbert空间且$\dim$ $\mathcal {H}$$\geqslant 2$,$\mathcal {B}$$_{s}($$\mathcal {H})$是$\mathcal {H}$上的所有自伴算子的全体.设$\varphi$是$\mathcal {B}$$_{s}(\mathcal {H})$上的线性满射,则$\varphi$保持Jordan积非零投影性当且仅当存在$\mathcal {H}$上的酉算子或反酉算子$U$以及常数$\lambda$满足$\lambda\in\{1,-1\}$,使得对任意的$X\in \mathcal {B}$$_{s}(\mathcal {H})$有$\varphi(X)=\lambda U^*XU$.

英文摘要:

      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 $\lambda\in\{1,-1\}.$