设H为无限维的复Hilbert空间, S(H)是H上全体对称算子构成的Jordan代数, Φ: S(H) → S(H)为双射且Φ(I)=I.证明下列条件等价: (1) Φ(ABA)=Φ(A)Φ(B)Φ(A),∀ A,B∈S(H); (2)Φ((1/2)(AB+BA))=(1/2)Φ(A)Φ(B)+(1/2)Φ(B)Φ(A),∀ A,B∈S(H); (3)Φ(ABC+CBA)=Φ(A)Φ(B)Φ(C) +Φ(C)Φ(B)Φ(A),∀ A,B,C∈S(H); (4)Φ((1/2)(ABC+CBA))=(1/2)Φ(A)Φ(B)Φ(C)+(1/2)Φ(C)Φ(B)Φ(A), ∀ A,B,C∈S(H); (5) Φ是S(H)上的Jordan环同构;(6)存在有界可逆的线性或共轭线性算子 A: H → H,At=A-1,使得 Φ(X)=AXAt, ∀ X∈ S(H). 得到了S(H)上Jordan环同构的新刻画.
Abstract
Let H be an infinite dimensional complex Hilbert space and S(H) be the Jordan algebra of all symmetric operators in B(H). We show that if bijective maps Φ : S(H) → S(H) with Φ(I) = I, then the following conditions are equivalent: (1) Φ(ABA) = Φ(A)Φ(B)Φ(A), ∀ A,B ∈ S(H); (2) Φ((1/2)(AB + BA)) = (1/2)Φ(A)Φ(B)+ (1/2)Φ(B)Φ(A), ∀ A,B ∈ S(H); (3) Φ(ABC +CBA) = Φ(A)Φ(B)Φ(C)+ Φ(C)Φ(B)Φ(A), ∀ A,B,C ∈ S(H); (4) Φ((1/2)(ABC + CBA)) = 1 2Φ(A)Φ(B)Φ(C) + (1/2)Φ(C)Φ(B)Φ(A), ∀ A,B,C ∈ S(H); (5) Φ is a Jordan ring isomorphism on S(H); (6) there exists a bounded invertible linear or conjugate linear operator A : H → H with At = A-1 such that Φ(X) = AXAt for every X ∈ S(H). New characterizations of Jordan ring isomorphism on S(H) were got.
关键词
对称算子 /
就范正交基 /
Jordan环同构
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Key words
symmetric operators /
orthonormal basis /
Jordan ring somorphism
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参考文献
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脚注
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基金
国家自然科学基金资助项目(11001194);山西省强校工程人才支持计划资助项目(TYAL)
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