摘要
设B(H)是维数大于1的复Hilbert空间H上有界线性算子全体得到的代数.■A,B∈B(H),定义拟积A。B=A+B-AB.证明Ф是B(H)上的双射且满足Ф(A^*。B)=Ф(A)^*。Ф(B),■A,B∈B(H)的充要条件是当dim H≥3时,存在H上的酉算子或共轭酉算子U使得Ф(A)=UAU^*,A∈B(H);当dim H=2时,存在H上的酉算子U使得Ф(A)=UAτU^*,A∈B(H),其中τ是C上的环自同构.设A=(aij)∈M2,则令Aτ=τ(a ij).
Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H≥2.For any A,B∈B(H),define the quasi-product of A and B as A B=A+B-AB.It is proved that a bijection Ф on B(H)satisfysФ(A^*.B)=Ф(A)^*.Ф(B)for all A,B in B(H)if and only if there is a unitary or an anti-unitary operator U on H such that Ф(A)=UAU^* for all A in B(H).When dim H≥3 or there is a unitary operator U on H such that Ф(A)=UAτU^* for all A in B(H)when dim H=2,whereτis a ring automorphism on C and Aτ=τ(aij)for all A=(aij)in M 2.
作者
宋显花
SONG Xianhua(College of Mathematics and Statistics,Qinghai Normal University,Xining 810008,Qinghai)
出处
《四川师范大学学报（自然科学版）》
CAS
北大核心
2019年第5期667-673,共7页
Journal of Sichuan Normal University（Natural Science）
基金
青海省科技厅项目(2018-ZJ-925Q和2017-ZJ-790)
关键词
算子代数
保持
拟积
同构
operators algebra
preserve
quasi-product
isomorphism
