Let N and M be nests on Banach spaces X and Y over the real or complex field F,respectively,with the property that if M∈M such that M-=M,then M is complemented in Y.Let AlgN and AlgM be the associated nest algebras.A...Let N and M be nests on Banach spaces X and Y over the real or complex field F,respectively,with the property that if M∈M such that M-=M,then M is complemented in Y.Let AlgN and AlgM be the associated nest algebras.Assume that Φ:AlgN→AlgM is a bijective map.It is proved that,if dim X=∞ and if there is a nontrivial element in N which is complemented in X,then Φ is Lie multiplicative (i.e.Φ([A,B])=[Φ(A),Φ(B)] for all A,B∈AlgN) if and only if Φ has the form Φ(A)=-TA*T-1+τ(A) for all A∈AlgN or Φ(A)=TAT-1+τ(A) for all A∈AlgN,where T is an invertible linear or conjugate linear operator and τ:AlgN→FI is a map with τ([A,B])=0 for all A,B∈AlgN.The Lie multiplicative maps are also characterized for the case dim X<∞.展开更多
Let H and K be indefinite inner product spaces. This paper shows that a bijective map Φ: B(H) → B(K) satisfies Φ(AB+ + B+A) = Φ(A)Φ(B)+ + Φ(B)+Φ(A) for every pair A,B ∈ B(H) if and only if either Φ(A) = cUAU+...Let H and K be indefinite inner product spaces. This paper shows that a bijective map Φ: B(H) → B(K) satisfies Φ(AB+ + B+A) = Φ(A)Φ(B)+ + Φ(B)+Φ(A) for every pair A,B ∈ B(H) if and only if either Φ(A) = cUAU+ for all A or Φ(A) = cUA+U+ for all A; Φ satisfies Φ(AB+A) = Φ(A)Φ(B)+Φ(A) for every pair A, B ∈ B(H) if and only if either Φ(A) = UAV for all A or Φ(A) = UA+V for all A, where A+ denotes the indefinite conjugate of A, U and V are bounded invertible linear or conjugate linear operators with U+U = c-1I and V+V = cI for some nonzero real number c.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10871111)Tian Yuan Foundation of China (Grant No. 11026161)Foundation of Shanxi University
文摘Let N and M be nests on Banach spaces X and Y over the real or complex field F,respectively,with the property that if M∈M such that M-=M,then M is complemented in Y.Let AlgN and AlgM be the associated nest algebras.Assume that Φ:AlgN→AlgM is a bijective map.It is proved that,if dim X=∞ and if there is a nontrivial element in N which is complemented in X,then Φ is Lie multiplicative (i.e.Φ([A,B])=[Φ(A),Φ(B)] for all A,B∈AlgN) if and only if Φ has the form Φ(A)=-TA*T-1+τ(A) for all A∈AlgN or Φ(A)=TAT-1+τ(A) for all A∈AlgN,where T is an invertible linear or conjugate linear operator and τ:AlgN→FI is a map with τ([A,B])=0 for all A,B∈AlgN.The Lie multiplicative maps are also characterized for the case dim X<∞.
基金Project supported by the National Natural Science Foundation of China (No.10471082) the Shanxi Provincial Natural Science Foundation of China (No.20021005).
文摘Let H and K be indefinite inner product spaces. This paper shows that a bijective map Φ: B(H) → B(K) satisfies Φ(AB+ + B+A) = Φ(A)Φ(B)+ + Φ(B)+Φ(A) for every pair A,B ∈ B(H) if and only if either Φ(A) = cUAU+ for all A or Φ(A) = cUA+U+ for all A; Φ satisfies Φ(AB+A) = Φ(A)Φ(B)+Φ(A) for every pair A, B ∈ B(H) if and only if either Φ(A) = UAV for all A or Φ(A) = UA+V for all A, where A+ denotes the indefinite conjugate of A, U and V are bounded invertible linear or conjugate linear operators with U+U = c-1I and V+V = cI for some nonzero real number c.
