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 Bs(H) the space of all self-adjoint operators in B(H). A characterization is given for additive bijective map Φ on Bs(H) satisfying [Φ(A2),Φ(A)]=0 for all ABs(H). It is showed that, if Φ(Fs(H))RI or  RIΦ(RI), then Φ has the form Φ(A)=cUAU+f(A)I, ABs(H), where cR, c0, U:HH is an unitary or conjugate unitary operator, and f is an additive real functional of Bs(H).

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

CLC Number: 

  • 47B49
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