| Additive Maps Preserving Similarity of Operators on Banach Spaces |
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| 作者姓名： | Jin Chuan HOU Xiu Ling ZHANG |
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| 作者单位： | Department of Mathematics, Shanxi Teachers University, Linfen 041004, P, R. China |
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| 基金项目： | This work is supported by NNSFC and PNSFS |
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| 摘 要： | Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.
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| 关 键 词： | 加法映射 算子代数 相似性 Banach空间 Hilbert空间 |
| 收稿时间： | 2003-07-07 |
| 修稿时间： | 2003-07-072004-02-11 |
Additive Maps Preserving Similarity of Operators on Banach Spaces |
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| Jin Chuan HOU Xiu Ling ZHANG.Additive Maps Preserving Similarity of Operators on Banach Spaces[J].Acta Mathematica Sinica,2006,22(1):179-186. |
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| Authors: | Jin Chuan Hou Xiu Ling Zhang |
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| Institution: | (1) Department of Mathematics, Shanxi Teachers University, Linfen 041004, P. R. China |
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| Abstract: | Let X be an infinite–-dimensional complex Banach space and denote by ℬ(X) the algebra of
all bounded linear operators acting on X. It is shown that a surjective additive map Φ from ℬ(X) onto
itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible
linear or conjugate linear operator A and a similarity invariant additive functional φ on ℬ(X) such
that either Φ(T) = cATA
−1 + φ(T)I for all T, or Φ(T) = cAT*A
−1 + φ(T)I for all T. In the case
where X has infinite multiplicity, in particular, when X is an infinite–dimensional Hilbert space, the
above similarity invariant additive functional φ is always zero.
This work is supported by NNSFC and PNSFS |
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| Keywords: | Additive maps on operator algebras Similarity of operators Banach spaces |
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