Generalized Jordan Derivation And Generalized Antiderivation On A Triangular Matrix Algebra

The study of operator algebra theory began in 1930s. With thefast development of the theory, now it has become a hot branch playing the roleof an initiator in modern mathematics. It has unexpected relations and mutualpenetration with quantum mechanics, noncommutative geometry, linear system andcontrol theory, indeed number theory as well as some other important branches ofmathematics. In order to discuss the structure of operator algebras, in recent years,a lot of scholars both here and abroad have focused on linear mappings of operatoralgebras and have been introduced more and more new methods. For instance, localmappings, bilocal mappings, linear preserving problems and so on were introducedsuccessively, at present time these mappings have become important tool in studyingoperator algebras. Among all kinds of operator algebras, nest algebras is a kindof very important non-selfadjoint and non-semiprime algebra. When the algebra isinfinity dimention, it is very complicated. The model of finity dimentional is trianglematrix algebra. In this paper, we mainly dicuss some characteristics of generalizedJordan derivation, generalized derivation and generalized antiderivation. The detailsas following:In Chapter 1, some notations, definitions are introduced and some theoremsare given. In section 2, some concopts are introduced, such as the definition ofgeneralized Jordan derivation, generalized derivation, prime and semiprime ring. Insection 3, we give a given theorem that we Will use in this paper.In Chapter 2, we first discuss a linear mapping on semiprime normed complex~*-algebra, and we have proved that if a linear mappingδfrom A into its bimodulesatisfiedδ(p)= pδ(p)+δ(p)p-pδ(I)p, for all p∈P_A, thenδ(ω) =ωδ(ω)+δ(ω)ω-ωδ(I)ω, for allω∈D_A. Moreover, if D_A dense in H_A, andδis continuous, thenδisa generalized derivation. Moreover, we proved that a generalized Jordan derivationfrom nonconmunity simple ring with identity operator I into its 2-torsion free and3-torsion free bimodule is a generalized derivation.In Chapter 3, we discuss generalized Jordan derivation from 2-torsion free tri-angular matrix into its 2-torsion free bimodule is the sum of a generalized derivaton and an antiderivation, and proved that if we define antidervation is zero on diagonalmatrics, then the factorzation is unquieness.In Chapter 4, we first present the notion of generalized antiderivation anddiscuss some result of generalized antiderivation, and prove that there is no propergeneralized antiderivation from upper triangular matrix algebra into its some matrixbimodule.