Characterization Of Jordan Derivations And Centralizers On Operator Algebras

As is well known, Jordan derivations and centralizers are very important maps in operator algebras and operator theory, and have received a fair amount of attention. In this paper, we will give new characterizations of Jordan derivations and centralizers on some operator algebras.We obtain the following results:1. Under some mild conditon, we shoow that nonlinear Jordan derivations on some operator algebras are automatically additive. Let A be a ring with a nontrivial idempotent P and δ be a map from A into its bimodule M. If A satisfies:(1) for M∈M, PMPA(I-P)= 0=> PMP= 0; (2) for M ∈M, PA(I-P)(I-P)M(I-P)= 0(?)(I-P)M(I-P)= 0, then δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) for all A, B∈A if and only if it is an additive derivation.2. We give a sufficient and necessary condition for additive maps on some subspace lattice algebras which are Jordan derivable at an arbitrary but fixed point. Let L be a subspace lattice on a Hilbert space H and AlgL be the associated subspace lattice algebra. Assume Ω∈AlgL and 8:Alg→AlgL is a linear map, if δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) for any A, B∈Alg L with AB=Ω, we say δ is Jordan derivable at Ω. In this chapter, we give a necessary and sufficient condition for a linear map which is Jordan derivable at Ω. In particular, we obtain that a linear map from a nest algebra into itself which is Jordan derivable at any point if and only if it is a derivation.3. We characterize a kind of linear maps on matrix algebras by cube-zero elements. LetF be a field of characteristic not 2 and f:Mn(F)→Mn(F) be a linear map. If xf(x)= 0 for any x∈Mn(F) with x3= 0, then there is a a, b∈Mn(F) such that f(x)= xa+trace(x)b,(?)xe∈ Mn(F).4. Equivalent characterizations of entralizers on B(H) are obtained. Let H be an infinite dimensional Hilbert space,if an additive map Φ:B(H)→B(H)satisfies Φ(A)A=AΦ(A)= Φ(A2)for any A∈B(H)with A2=0,then Φ(A):Φ(I)A=AΦ(I) for any A∈B(H),that is Φ is a centralizer on B(H).