Operator Algebras Of ¦Î-lie Derivative Mapping

The study of operator algebra theory began in 30times of the 20th century. With the fast development of the theory, now it has become an important area in modern mathematics. And triangular algebra is a class of most important nonprime and non-semisimple operator algebras. On the basis of existing papers, in this paper we mainly and detailedly discuss (generalized)ξ-Lie derivations on B(X), (generalized)ξ-Lie derivable maps at zero point of triangular algebras. The details as following:In chapter 1, some notions, definitions (for example, triangular algebra, nest algebra, derivation, Lie derivation and so on) and some well-known theorems are given.In chapter 2, we mainly characterizeξ-Lie derivations on B(X). We prove that the linear mapδsatisfyingδ([A,B]ξ)= [δ(A),B]ξ+[A,δ(B)]ξ(ξ≠0,±1) with AB= 0 (or AB is a fixed nontrivial idempotent) on B(X) is of the form A→AT-TA (T∈B(X)). We also discuss generalizedξ-Lie derivations on B(X).In chapter 3, we first prove thatξ-Lie derivable mapδat zero point of triangular algebra is of the form T→d(T)+δ(I)T forξ≠1, where d is a additive derivation. As applications,ξ-Lie derivable maps at zero point on upper triangular block matrix algebras and Banach space nest algebras are characterized. Subsequently, we prove that the structure of generalizedξ-Lie derivable maps at zero point of triangular algebras is the same asξ-Lie derivable maps at zero point.