Additive Maps Preserving Nilpotent Perturbation of Scalars |
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Authors: | Zhang Ting Hou Jin Chuan |
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Affiliation: | 1. School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P. R. China;2. Department of Mathematics, Shanxi University, Taiyuan 030006, P. R. China |
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Abstract: | Let X be a Banach space over F (=R or C) with dimension greater than 2. Let N (X) be the set of all nilpotent operators and B0(X) the set spanned by N (X). We give a structure result to the additive maps on FI + B0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T)=cATA-1 +φ(T)I for all T ∈ FI + B0(X) or Φ(T)=cAT* A-1 + φ(T)I for all T ∈ FI + B0(X), where c is a nonzero scalar, A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional. In addition, if dim X=∞, then A is in fact a linear or conjugate linear invertible bounded operator. |
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Keywords: | Banach spaces nilpotent operators perturbations additive preservers |
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