Maps preserving the idempotency of products of operators
Maps preserving the idempotency of products of operators
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DOI：
10.1016/j.laa.2006.11.013
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发表时间：
2007-07
影响因子：
1.1
通讯作者：
Li Fang;Guoxing Ji;Yongfeng Pang
中科院分区：
文献类型：
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作者：
Li Fang;Guoxing Ji;Yongfeng Pang
设B（X）是Banach空间X上所有有界线性算子的代数，N（X）是B（X）中的幂零算子集.设满射A：B（X）→B（X）使得A，B∈B（X）满足AB∈N（X）当且仅当B（A）∈N（X）.若X是无限维的，则存在映射f：B（X）→C <${0}使得下列之一成立：若X有维数n且3 <$n<∞，且B（X）与n×n复矩阵的代数Mn相同，则存在映射f：Mn→C <${0}，域自同构<$：C→C，可逆S∈ Mn使得<$具有下列形式之一：其中At表示A的转置。将结果推广到两个以上算子的乘积和B（X）上的其它类型的乘积，包括Jordan三重积A B=阿坝.此外，在有限维情形下的结果被用来刻画矩阵上的满射映射保持矩阵乘积的谱半径。
Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ϕ:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ϕ(A)ϕ(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C⧹{0} such that one of the following holds: If X has dimension n with 3⩽n<∞, and B(X) is identified with the algebra Mnof n×n complex matrices, then there exist a map f:Mn→C⧹{0}, a field automorphism ξ:C→C, and an invertible S∈Mnsuch that ϕ has one of the following forms:where Atdenotes the transpose of A. The results are extended to the product of more than two operators and to other types of products on B(X) including the Jordan triple product A∗B=ABA. Furthermore, the results in the finite dimensional case are used to characterize surjective maps on matrices preserving the spectral radius of products of matrices.