域上2 × 2对称矩阵空间的加法秩保持(英文)

Additive preservers of rank on the spaces of 2 × 2 symmetric matrices over fields

令F是一个域,n是一个正整数.Sn(F)记F上所有n × n对称矩阵的集合.若一个算子f : Sn(F) → Sn(F)满足对任意的A,B ∈ Sn(F)都有f(A + B) = f(A) + f(B),则称之为加法的;若对任意的X ∈ Sn(F)都有rankf(X) =rankX,则称f为Sn(F)上的秩保持.当n ≥ 3及F为任意域时,Sn(F)上的所有加法秩保持已被作者在[4]中确定.这里,对于任意的F,S2(F)上所有的满足对每个X ∈ S2(F) \ {xD12|x ∈ F\{0}}都有rankf(X) =rankX的加法算子的一般形式被确定,由此S2(F)上的所有加法秩保持被刻划.

Let F be a ?eld and n be a positive integer. Denote the set of all n × n symmetric matrices over F by Sn(F). An operator f : Sn(F) → Sn(F) is said to be additive if f(A + B) = f(A) + f(B) for any A,B ∈ Sn(F), and is called a preserver of rank on Sn(F) if rankf(X) =rankX for every X ∈ Sn(F). When n ≥ 3, for any F the forms of the additive preservers of rank on Sn(F) have been determined by the author in [4]. Here, when F is arbitrary, the general form of all additive operators from S2(F) to itself satisfying rankf(X) =rankX for every X ∈ S2(F) \ {xD12|x ∈ F \ {0}} is determined. Thereby, the additive preservers of rank on S2(F) for any F are characterized.