Equivalent Characterization Of Centralizers On Operator Algebras

As is well known, left (right) centralizers and centralizers are very important maps in operator algebras and operator theory, and have received a fair amount of attention. In this paper, we characterize additive maps on triangular algebras and β(H) which are centralized (left or right centralized) at one given point, and establish conditions additive maps to be cen-tralizers (left or right centralizers), and obtain new equivalent characterization of centralizers (left or right centralizers). Main results of this paper are as follows.1. Equivalent characterization of left (right) centralizers on triangular algebras was got. Assume T= Tri(A,M,B) is a triangular algebra, ∈ T is an element such that A0 and B0 are right (left)-invertible elements of A and B respectively and M0 ∈ M is an arbitrary but fixed point. If for any A ∈ A, B ∈ B, there is n such that nI1- A is invertible in A and nI2 - B is invertible in B, then a additive map Φ:T → T satisfies Φ(ST)=Φ(S)T (Φ(ST)= SΦ(T)) for S, T ∈ T with ST= Ω if and only ifΦ(ST)= Φ(S)T (Φ(ST)= SΦ(T)), VS, T ∈ T, that is,Φ is a left (right) centralizer.2. We obtain new equivalent characterization of centralizers on triangular algebras (β(H), AlgN). Assume A is a triangular algebra (α(H), AlgN), and ΩA is an arbi-trary but fixed element. Then a additive map Φ:.A →A such that Φ(AB)=Φ(A)B= AΦ(B), A, B∈A, AB= Ω if and only ifΦ(AB)=Φ(A)B= AΦ(B), (?)A, B ∈ A, that is, Φ is a centralizer.3. We characterize left (right) centralizers on B(H) by square zero operators and invo-lutions separately. Let H be an infinite dimensional Hilbert space, and let Φ:β(H) →β(H) be an additive map. Then(1)Φ(A2)= AΦ(A) for A ∈B(H) with A2=0 (I) if and only if Φ(A)= AΦ(I) for all A ∈ B(H).(2)Φ(A2)=Φ(A)A for A∈β(H)with A2=0(I)if and only ifΦ(A)=Φ(I)A for all A∈β(H).(3)Φ(A2)=AΦ(A)=Φ(A)A for A∈β(H)with A.=0(I)if and only ifΦ(A)= AΦ(I)=Φ(I)A for all A∈β(H).