有限维张量积空间上的强可分算子

设F是一个域,a∈F~nF~m.若存在h∈F~m,k∈F~m,使得a=hk,则称a是可分的.空间F~nF~m上的线性算子A称为是强可分的,是指x∈F~nF~m,x可分Ax可分.本文证明了F~nF~n上的线性算子A是强可分的当且仅当存在F~n上的线性双射A_1与A_2,使得A=A_1A_2或A=A_1~T A_2;证明了F~nF~m(n≠m)上线性算子A是强可分的当且仅当存在F~n与F~m上的线性双射A_1与A_2,使得A=A_1A_2.最后,给出了可分算子、强可分算子和秩1保持映射之间的关系.

Strongly Separable Operators on Finite Dimensional Tensor Product Spaces

Let F be a field.An element a in F~n F~m is said to be separable if a = h k for some h∈F~n,k∈G F~m.A linear orperator A on F~n F~m is said to be strongly separable if x∈F~nF~m is separable if and only if Ax is separable.It is proved that a linear operator A on F~nF~n is strongly separable if and only if A=A_1A_2 or A = A_1~T A_2 for some linear bijections A_1,A_2 on F~n.Next,it is shown that when m≠n,a linear operator A on F~n F~m is strongly separable if and only if A = A_1A_2 for some linear bijections A_1,A_2 on F~n,F~m,respectively.Lastly,relationships among separable operators,strongly separable operators and rank-1 preservers are discussed.