Research Of Centralizers And High Jordan Derivations On Operator Algebras

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.