保持算子＋-乘积幂等性的映射

Mappings Preserving the Idempotency of ＋-Product Pooduct of Operators

设H为复的无限维的完备的不定内积空间,B(H)表示H上所有有界线性算子构成的代数.令A是B(H)中到少包含单位I和一秩幂等元的非零数乘C*I1(H)的子集,且对任意的A∈A,Gcv{A,I}■A.如果对任意的A,B∈A,AB+为非零幂等元当且仅当Φ(A)Φ(B)+为非零幂等元,则称Φ为A上保持算子+-乘积幂等性的映射。A上保持算子+乘积幂等性映射的具体形式得到了完整的刻画.当H为Hilbert空间时,作为推论,给出了A上保持算子*乘积幂等性的映射的具体形式.

Let H be an infinite dimensional complex completely indefinite inner product spaces,B（H） be the sets of all lincar operators on H. A be a subset of B（H）,which containing at least all nonzero scalar multiples of rank-one indempotents and I, and Gev{A, I}∈A for any A ∈ A. We obtain a general form of a surjective unital mapping Φ on A,whieh preserve the idempoten- cy of t-product of operators in both directions,i, e：AB＋ is a nonzero idempotent if and only if （A）Φ（B）＋ is a nonzer idempotent for any A,B∈A. As corollaries,the maps preserving the idemp- otency of ＊-product of operators on A are completely classified when H is a Hillbert spaces.