Keep The Operator Product Of Idempotent And Nilpotent Of The Linear Mapping

The aim of linear preserver problem is to character linear maps on operator algebras preserving certain properties,subsets or relations.And the research of this problem has yield fruitful results.At present,this issue become increasingly attracted more and more people's attention.Recently,some preserver problems concerning certain properties of the products of operators have been considered (cf[1,2]).In this paper we mainly discuss the linear maps preservering idempotency and nipotency of products of operators.The main results are as follows.1.Letφbe a linear map on M_n.Thenφpreserves the nonzero idempotency of products of two operators if and only if there exists an invertible matrix A∈M_n and a constantλ∈{1,-1} such that one of the following forms holds.(1)φ(X) =λAXA^-1 for all X∈M_n;(2)n=2 andφ(X) =λAX^tA^-1 for all X∈M_n,where X^t denotes the transpose of X.2.Let X be a complex infinite Banach space and letφbe a linear surjective map on B(X).Thenφpreserves the nonzero idempotency of products of two operators if and only if there exists an invertible operator A∈B(X) and a constantλ∈{1,-1} such thatφ(X)=λAXA^-1 for all X∈B(X).3.Letφbe a linear map on M_n.Thenφpreserves the nonzero idempotency of triple Jordan products of two operators if and only if there exists an invertible matrix A∈M_n and a constantεwithε³=1 such that one of the following forms holds.(1)φ(X)=εAXA^-1 for all X∈M_n;(2)φ(X)=εAX^tA^-1 for all X∈M_n,where X^t denotes the transpose of X.4.Let X be a complex infinite dimensional Banach space and letφbe a linear surjective map on B(X).Thenφpreserves the nonzero idempotency of triple Jordan products of two operators if and only if there exists a constantεwithε³=1 such that one of the following forms holds.(1)There is an invertible operator A∈B(X) such thatφ(X)=εAXA^-1 for all x∈B(X);(2)X is reflexive and there is an invertible operator A from X′onto X such thatφ(X)=εAX′A^-1 for all X∈B(X),where X′denotes the adjiont of X. 5.Letφ:B(H)â†'B(K) be a linear surjective map withφ(I)≠0.Thenφpreserves the nilpotency of trpile Jordan products of two operators if and only if there exists a nonzero complex constant c and an invertible operator A∈B(H,K) such that one of the following forms holds.(1)φ(T)=cATA^-1 for all T∈B(H);(2)φ(T)=cAT^trA^-1 for all T∈B(H),where T^tr denotes the transpose of T relative to a fixed but arbitrary orthonormal base of H.