| Additive Maps Preserving Nilpotent Operators or Spectral Radius |
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| 作者姓名： | Zhao Fang BAI Jin Chuan HOU |
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| 作者单位： | [1]School of Science, Xi＇an Jiaotong University, Xi＇an 710049, P. R. China [2]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 a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.
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| 关 键 词： | 附加保护 幂算子 光谱范围 Banach空间 子空间 |
| 收稿时间： | 2003-06-04 |
| 修稿时间： | 2003-06-042003-08-06 |
Additive Maps Preserving Nilpotent Operators or Spectral Radius |
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| Zhao Fang BAI Jin Chuan HOU.Additive Maps Preserving Nilpotent Operators or Spectral Radius[J].Acta Mathematica Sinica,2005,21(5):1167-1182. |
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| Authors: | Zhao Fang Bai Jin Chuan Hou |
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| Institution: | (1) School of Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China;(2) Department of Mathematics, Shanxi Teachers University, Linfen 041004, P. R. China;(3) Department of Mathematics, Shanxi Teachers University, Linfen 041004, P. R. China |
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| Abstract: | Let X be a (real or complex) Banach space with dimension greater than 2 and let ℬ0(X)
be the subspace of ℬ(X) spanned by all nilpotent operators on X. We get a complete classification
of surjective additive maps Φ on ℬ0(X) which preserve nilpotent operators in both directions. In
particular, if X is infinite–dimensional, we prove that Φ has the form either Φ(T) = cATA
−1 or
Φ(T) = cAT'A
−1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar,
T' denotes the adjoint of T. As an application of these results, we show that every additive surjective
map on ℬ(X) preserving spectral radius has a similar form to the above with |c| = 1.
This work is supported by NNSFC and PNSFS |
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| Keywords: | Additive preservers Nilpotent operators Spectral radius |
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