Set The Centralizer Algebra And Higher Order Lie Derivations

The study of non-selfadjoint operator algebra theory began in 60times years of 20 century. With the fast development of the theory, now it has become an important area in studying operator algebras. And nest algebra is a class of most important nonprime and non-selfadjoint operator algebras. On the basis of existing papers, in this paper we mainly and detailedly discuss centralizers, all-derivable points and nonlinear higher Lie derivations on nest algebras. The details as following:In chapter 1, we give some notions, definitions (for example, triangular alge-bra, nest algebra, centralizer, all-derivable point and so on) and some well-known theorems.In chapter 2, we mainly discuss centralizers on nest algebras. We prove that the additive mapφsatisfying (m+n)φ(Ap+1) - mφ(A)Ap - nApφ(A)∈FI orφ(Am+n+1) - Amφ(A)An∈FI (m,n are positive integers) on nest algebra is of the form Aâ†'λA (λ∈F).In chapter 3, we first discuss all-derivable points of triangular algebras. We prove that are all-derivable points of trian-gular algebras. As an application, we give some all-derivable points on nest algebras without the assumption of strong operator topology continuity. Subsequently, we discuss nonlinear higher Lie derivations on nest algebras and we prove that every nonlinear higher Lie derivation D=(Li)i∈N from T(N) into itself is of the form Ln(A)=δn(A)+hn(A)I for all A∈T(N) and all n∈N, where (δi)i∈N is an additive higher derivation and (hi)i∈N is a sequence of nonlinear functionals with hn([A,B])=0 for all A, B∈T(N) and all n∈N.