Characteristics Of Jordan Derivations And Lie Derivations On Operator Algebras

In recent years, the questions of characterizingξ-Lie derivations and revealing the relationships ofξ-Lie derivations have received many mathematicians'attention and interest. There have been a number of papers on the study. For example, Zhu in [1] showed that:(1) every invertible operator in nest algebras is an all-derivable point for the strong operator topology; (2) Zhu, Xiong and Zhang in [2] showed that every matrix G≠0is an all derivable point of the algebra of n×n upper triangular matrices. Cheung in [3] gave sufficient conditions such that every Lie derivation on such an algebraÏ…is a sum of derivation onÏ…and a mapping fromÏ…to its centre. Lu and Jing in [4] proved that ifδ: B ( X)→B(X) is a linear map satisfyingδ([ A,B]) = [δ(A) ,B ]+[A,δ(B)] for any A, B∈B(X) with AB =0 (resp. AB = P, where P is a fixed nontrivial idempotent), thenδ= d +Ï„, where d is a derivation of B (X) andÏ„: B ( X)→CI is a linear map vanishing at commutators [ A,B ] with AB =0 (resp. AB = P). Zhang in [5] proved that every Jordan derivation of triangular algebras is a derivation. Lu in [6] showed that a continuous linear mapδfrom Banach algebras into its Banach bimodule which satisfies the derivation equation when AB is a fixed left (or right) invertible element is a Jordan derivation and a linear map from Banach algebra into its bimodule satisfying the derivation equation when AB is a fixed idempotent which is faithful is a derivation.Recently, Zhu, Xiong and Zhang in [7] showed that every matrix G≠0is an all derivable point of the algebra of n×n matrices. Qi and Hou in [8] proved that: (1) an additive map L is an additive (generalized) Lie derivation if and only if it is the sum of an additive (generalized) derivation and an additive map from the algebra into its center vanishing all commutators; (2) an additive map L is an additive (generalized)ξ-Lie derivation withξ≠1 if and only if it is an additive (generalized) derivation satisfying L (ξA )=ξL(A) for all A . Xiao and Wei in [9] showed that any Jordan higher derivation on triangular algebra is a higher derivation.Motivated by those results, we get the following results in the text. This dissertation is divided into five chapters. In the first chapter, it presents the related symbols, definitions and the background of the research. Finally, it presents the content and the meaning of the research. In the second chapter, motivated by Zhu, Xiong and Zhang's paper (see [7]), we give the first definitions of the Jordan derivable mapping and the Jordan all-derivable point and get that every matrix (G|⁾ is a Jordan all-derivable point in M_K ( 2≤K≤n) if and only if every invertible matrix G₁ in M_K ( 2≤K≤n) is a Jordan all-derivable point. In the third chapter, motivated by Qi and Hou's paper (see [8]), we apply the second definitions of the Jordan derivable mapping and the Jordan all-derivable point and get that ifδ: U→U is aξ-Lie derivable mapping at G =(?), whenξ=1,δis the form ofδ₁ +Ï„, whereδ₁: U→U is a derivation andÏ„: U→C is a linear functional; whenξ≠1,δis a derivation. In the forth chapter, motivated by Xiao and Wei's paper (see [9]), we get that if X in (?) is a left (or right ) invertible operator and {δ_n}:(?)→M ' be linear maps which satisfyδ_n ( AB )(?)δ_i(A)δ_j(B) for all A,B∈(?) with AB = X, then {δ_n} are Jordan higher derivations. In the fifth chapter, we summarize the research and look forward to the future of some open problems about Jordan derivations and Lie derivations.