Acta mathematica scientia,Series A ›› 2010, Vol. 30 ›› Issue (6): 1686-1692.

• Articles • Previous Articles     Next Articles

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

 QI Xiao-Fei1,2, DU Quan-Ping2,4, HOU Jin-Chuan1,3   

  1. 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

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

CLC Number: 

  • 47B49
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