【作者】 马飞；

【导师】 张建华；

【作者基本信息】 陕西师范大学 ， 基础数学， 2013， 博士

【摘要】 摘要本文主要讨论的是算子代数上的可加及线性映射.运用算子代数的结构性质及代数分解的方法,研究了算子代数上的保持映射和高阶JOrdan导子,内容涉及标准算子代数上的中心化映射,三角代数上的中心化映射,CDC代数上的中心化子,自反代数上的中心化子及三角代数上的高阶JOrdan导子和三角代数上的高阶Jordan三重导子.全文共分五章,主要内容如下：第一章介绍了本文主要内容的研究背景,意义和现状,并列出了本文要用到的符号,介绍了本文后几章将用到的中心化子、导子、高阶导子等概念及本文的主要结论.第二章研究了标准算子代数上的中心化映射.首先讨论了标准算子代数上满足(m+n)Φ(Ar+1)-mΦ(A)Ar-nArΦ(A)∈(?)I(m,n,r为正整数)的可加映射Φ具有Φ(A)=λA(λ∈(?))的形式,然后讨论了标准算子代数上满足(m+n)Φ(ABA)-(mΦ(A)BA+nABΦ(A))∈(?)I(m,n为正整数)的可加映射Φ亦具有Φ(A)=λA(λ∈(?))的形式,并得到了在标准算子代数上的一些可加映射的等价刻画.第三章首先研究了三角代数上满足(m+n)Φ（A2)-（mΦ(A)A+nAΦ(A))∈(?)((?))(m,n∈N+)的可加映射,证明了其具有Φ(A)=λA(λ∈(?)((?))的形式.其次刻画了三角代数上保持(m+n)Φ(Ar+1)-(mΦ(A)Ar+nArΦ(A))∈(?)((?))(m,n,r∈N+)的可加映射Φ亦具有Φ(A)=λA(λ∈(?)((?))的形式.第四章首先研究了不可约CDC代数上满足(m+n)Φ(Ar+1)=mΦ(A)Ar+nArΦ(A)(m,n,r为正整数)的可加映射具有Φ(A)=λA(λ∈(?))的形式,进而研究了在任意的CDC代数上满足(m+n)Φ(Ar+1)=(mΦ(A)Ar+nArΦ(A)(m,n,r∈N+)的可加映射Φ亦是中心化子.另外利用自反代数的结构特征,证明了在自反代数上满足(m+n)Φ(Ar+1)=(mΦ(A)Ar+nArΦ(A)和Φ(Am+n+1)=AmΦ(A)An(m,n,r∈N+)的可加映射Φ均具有Φ(A)=∈λA(λ∈(?))的形式.第五章研究了三角代数上的广义高阶Jordan导子和广义高阶Jordan三重导子.本章引入了广义高阶Jordan导子、广义高阶Jordan三重导子和广义高阶导子的概念,利用三角代数的结构性质和代数分解方法,得到三角代数上的广义高阶Jordan导子和广义高阶Jordan三重导子都是广义高阶导子的结论.更多还原

【Abstract】 Abstract In this paper, we studied the additive and linear mappings of operator alge-bras. Using the properties of algebras and decomposition of algebras on operator algebras, we discussed the centralizers and high Jordan derivations on some operator algebras. The content includes centralizing mappings on standard operator algebra, centralizing map-pings on triangular algebra, centralizers on CDC algebras and reflexive algebras, general-ized higher Jordan derivation and generalized higher Jordan triple derivation of triangular algebras. This paper is divided into five chapters, the main contents are followed:In Chapter1, we first introduced some research background and present situation on our paper and list some notations. Second, we gave the definitions of centralizers, derivations, high derivation and so on. At last, the main theorems of this paper are given.In Chapter2, we devoted to study the centralizing mappings on standard opera-tor algebras. Firstly, we discussed that the additive mappingФ on the standard opera-tor algebra satisfying (m+n)Φ(Ar+I)-mΦ(A)Ar-nArΦ(A)∈(?)I(m,n,r∈N+), has the form Φ(A)=λA(A∈(?)). Secondly, we obtained the additive mapping satisfy-ing (m+n)Φ(ABA)-(mΦ(A)BA+nABΦ(A))∈(?)I ((m, n∈N+) also has the form Ф(A)=λA(λ∈(?)). Thirdly, we got some equivalent characterizations of additive map-ping on standard operator algebra.In Chapter3, we firstly discussed the additive mapping Φ on the triangular algebra satisfying (m+n)Φ(A2)-(mΦ(A)A+nAΦ(A))∈(?)((?))(m, n∈N+) and proved that Ф has the form Φ(A)=λA(Aλ∈(?)((?)).Secondly, we obtained the additive mapping satisfying (m+n))Ф(Ar+1)-(mΦ(A)Ar+nArΦ(A))∈(?)((?))(m,n, r∈N+) also has the form Φ(A)=λA(λ∈(?)((?)).In Chapter4, we firstly got that the additive mappingФ on the irreducible completely decomposition CSL(CZDC) algebras satisfying (m+n)Ф(Ar+1)=mΦ(A)Ar+nArΦ(A)(m,n,r∈N+) has the form Φ(A)=λA（λ∈(?)). Secondly, we proved that the additive mappingΦ on any CDC algebras satisfying (m+n)Φ(Ar+1)=mΦ(A)Ar+nArΦ(A) also is a centralizr. At last, we proved that the additive mappingΦ on the reflexive algebras which satisfies (m+n)Φ(Ar+1)=mΦ(A)Ar+nArΦ(A) orΦ(Am+n+1)=AmФ(A)An (m, n, r∈N+) has the form Φ(A)=λA(λ∈(?)) by the structure properties of reflexive algebras.In Chapter5, we studied the generalized higher Jordan derivation and generalized higher Jordan triple derivation on the triangular algebras. We firstly introduce the defini-tion of generalized higher Jordan derivation, generalized higher Jordan triple derivation and generalized higher derivation. Secondly, using the structure properties of the triangu-lar algebras and decomposition of algebras, we got that both the generalized higher Jordan derivation and generalized higher Jordan triple derivation on the triangular algebras are generalized higher derivation.更多还原