自伴算子空间上满足[Φ(A2), Φ(A)] = 0的可加满射

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自伴算子空间上满足[Φ(A²), Φ(A)] = 0的可加满射

Additive Maps Satisfying [Φ(A²), Φ(A)] = 0 on the Space of Self-adjoint Operators

自伴算子空间上满足[Φ(A²), Φ(A)] = 0的可加满射

Additive Maps Satisfying [Φ(A²), Φ(A)] = 0 on the Space of Self-adjoint Operators

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数学物理学报 ›› 2010, Vol. 30 ›› Issue (6): 1686-1692.

齐霄霏^1,2|杜拴平^2,4|侯晋川^1,3

1.山西大学数学学院太原 030006|2.山西师范大学数计学院山西临汾 041004|3.太原理工大学数学系太原 030024; 4.厦门大学数学学院福建厦门 361005

收稿日期:2008-10-08 修回日期:2009-12-06 出版日期:2010-12-25 发布日期:2010-12-25
基金资助:
国家自然科学基金(10771157)、山西省自然科学基金(2006021008, 2007011016)和山西省回国留学人员研究基金(2007-38)资助

QI Xiao-Fei^1,2, DU Quan-Ping^2,4, HOU Jin-Chuan^1,3

1.Department of Mathematics, Shanxi University, Taiyuan 030006|2.Department of Mathematics, Shanxi Normal University, Shanxi Linfen 041004|3.Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024|4.Department of Mathematics, Xiamen University, Fujian Xiamen 361005

Received:2008-10-08 Revised:2009-12-06 Online:2010-12-25 Published:2010-12-25
Supported by:
国家自然科学基金(10771157)、山西省自然科学基金(2006021008, 2007011016)和山西省回国留学人员研究基金(2007-38)资助

摘要：

令H为维数大于2的复 Hilbert 空间, B}_{s}(H) $为$ H $上所有有界自伴算子构成的实线性空间.该文给出$ {\mathcal B}_{s}(H) $上满足$ [\Phi(A^2),\Phi(A)]=0 $对所有$ A\in {\mathcal B}_s(H) $成立的可加双射$ \Phi $的刻画,在$ \Phi({\mathcal F}_{s}(H)) \not\subseteq{\mathbb R}I $或$ {\mathbb R}I\subseteq\Phi({\mathbb R}I) $的条件下证明了上述$ \Phi $具有形式$ \Phi(A)=cUAU^*+f(A)I,\ \forall A \in {\mathcal B}_{s}(H), $其中$ c\in {\Bbb R}, c\neq 0 $,$ U:H\rightarrow H $是酉算子或共轭酉算子, 而$ f $是$ {\mathcal B}_{s}(H)$上的可加泛函.

关键词: 可加映射, 交换性, Jordan同态, 自伴算子空间

Abstract:

Let $H$ be a complex Hilbert space with dimension greater than 2 and ${\mathcal B}_{s}(H)$ the space of all self-adjoint operators in ${\mathcal B}(H)$ . A characterization is given for additive bijective map $\Phi$ on ${\mathcal B}_{s}(H)$ satisfying $[\Phi(A^2),\Phi(A)]=0$ for all $A\in {\mathcal B}_s(H)$ . It is showed that, if $\Phi({\mathcal F}_{s}(H)) \not\subseteq {\mathbb R}I$ or ${\mathbb R}I\subseteq\Phi({\mathbb R}I)$ , then $\Phi$ has the form $\Phi(A)=cUAU^*+f(A)I, ~ \forall A \in {\mathcal B}_{s}(H),$ where $c\in {\mathbb R},$ $c\neq 0$ , $U:H\rightarrow H$ is an unitary or conjugate unitary operator, and $f$ is an additive real functional of ${\mathcal B}_{s}(H)$ .

Key words: Additive maps, Commutativity, Jordan homomorphism, Space of self-adjoint operators

中图分类号:

47B49

齐霄霏, 杜拴平, 侯晋川. 自伴算子空间上满足[Φ(A²), Φ(A)] = 0的可加满射[J]. 数学物理学报, 2010, 30(6): 1686-1692.

QI Xiao-Fei, DU Quan-Ping, HOU Jin-Chuan. Additive Maps Satisfying [Φ(A²), Φ(A)] = 0 on the Space of Self-adjoint Operators[J]. Acta mathematica scientia,Series A, 2010, 30(6): 1686-1692.

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