Let H be a complex Hilbert space of dimension greater than 2, and B（H） denote the Banach algebra of all bounded linear operators on H. For A, B C B（H）, define the binary relation A ≤ B by A＊A = A＊B and AA＊ = A...Let H be a complex Hilbert space of dimension greater than 2, and B（H） denote the Banach algebra of all bounded linear operators on H. For A, B C B（H）, define the binary relation A ≤ B by A＊A = A＊B and AA＊ = AB＊. Then （B（H）, ＂〈.＂） is a partially ordered set and the relation ＂≤＂ is called the star order on B（H）. Denote by Bs（H） the set of all self-adjoint operators in B（H）. In this paper, we first characterize nonlinear continuous bijective maps on Bs（H） which preserve the star order in both directions. We characterize also additive maps （or linear maps） on B（H） （or nest algebras） which are multiplicative at some invertible operator.展开更多
Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a l...Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a linear bijective map ψA →A satisfying ψ（A）ψ（B） - ψ（B）ψ（A）＊ = AB - BA＊ for A, B ∈ A and a real functional h on A with h（0） -= 0 such that Ф（A） = ψ（A） ＋ h（A）I for every A ∈ A. In particular, if .4 is a type I factor, then, Ф（A） = cA ＋ h（A）I for every A ∈ .4, where c = ±1.展开更多
Let B（X） be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n 〉 1, we show that an additive surjective map Ф on B（X） preserves Drazin inver...Let B（X） be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n 〉 1, we show that an additive surjective map Ф on B（X） preserves Drazin invertible operators of index non-greater than n in both directions if and only if Ф is either of the form Ф（T） = aATA-1 or of the form Ф（T） = aBT＊B-1 where a is a non-zero scalar, A ： X → X and B ： X＊ → X are two bounded invertible linear or conjugate linear operators.展开更多
基金Supported by National Natural Science Foundation of China （Grant Nos. 10871111, 10501029） and the Specialized Research Fund for Doctoral Program of Higher Education （Grant No. 200800030059）
文摘Let H be a complex Hilbert space of dimension greater than 2, and B（H） denote the Banach algebra of all bounded linear operators on H. For A, B C B（H）, define the binary relation A ≤ B by A＊A = A＊B and AA＊ = AB＊. Then （B（H）, ＂〈.＂） is a partially ordered set and the relation ＂≤＂ is called the star order on B（H）. Denote by Bs（H） the set of all self-adjoint operators in B（H）. In this paper, we first characterize nonlinear continuous bijective maps on Bs（H） which preserve the star order in both directions. We characterize also additive maps （or linear maps） on B（H） （or nest algebras） which are multiplicative at some invertible operator.
基金supported by National Natural Science Foundation of China(10871111)the Specialized Research Fund for Doctoral Program of Higher Education(200800030059)(to Cui)Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education,Science and Technology(NRF-2009-0070788)(to Park)
文摘Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a linear bijective map ψA →A satisfying ψ（A）ψ（B） - ψ（B）ψ（A）＊ = AB - BA＊ for A, B ∈ A and a real functional h on A with h（0） -= 0 such that Ф（A） = ψ（A） ＋ h（A）I for every A ∈ A. In particular, if .4 is a type I factor, then, Ф（A） = cA ＋ h（A）I for every A ∈ .4, where c = ±1.
文摘Let B（X） be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n 〉 1, we show that an additive surjective map Ф on B（X） preserves Drazin invertible operators of index non-greater than n in both directions if and only if Ф is either of the form Ф（T） = aATA-1 or of the form Ф（T） = aBT＊B-1 where a is a non-zero scalar, A ： X → X and B ： X＊ → X are two bounded invertible linear or conjugate linear operators.
