Additive Maps Preserving Nilpotent Perturbation of Scalars  

作　　者：Ting ZHANG Jin Chuan HOU 

机构地区：[1]School of Mathematics, Taiyuan University of Technology [2]Department of Mathematics, Shanxi University

出　　处：《Acta Mathematica Sinica,English Series》2019年第3期407-426,共20页数学学报（英文版）

基　　金：Supported by Natural Science Foundation of China(Grant No.11671294)

摘　　要：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 B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A^(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A^(-1)+ φ(T)I for all T ∈ FI + B_0(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.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 B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A^(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A^(-1)+ φ(T)I for all T ∈ FI + B_0(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.

关 键 词：BANACH SPACES NILPOTENT operators PERTURBATIONS additive PRESERVERS 

分 类 号：O1[理学—数学]