Derivable Mappings On Triangular Algebras

The study of operator algebra theory began in 1930s.Though compary with some other theroy it is relatively new, but it has unexpected application in some mathematics theroy and other subjects, such as quantum mechanics,linear system , noncommutative geometry, contral theroy and some other important branches of mathematics. Accompany with its using in other subjects, this theory developed a lot.Now it has become a hot branch in modern mathematics. In order to dis-cuss the structure of operator algebras, in recent years, many scholars both her and abroad have focused on mappings on operator algebras. They have introduced some new concepts and new methods. For example, Jordan derivable map, generalized derivable mappings at zero, higher derivation, higher Jordan derivation etc. At present time these mappings have become important tools in studying operator al-gebras. Triangular algebra is a class of most important nonprime and nonself adjoint operator algebras. Such as upper triangular matrix algebra and nest algebra are this algebra. On the basis of existing papers, in this paper we mainly and detailedly discuss generalized higher Jordan derivations, generalized derivable mappings at zero and Jordan derivable mappings at unit on Triangular algebra. The details as following:In chapter 1, we give some notions, definitions (for example, triangular algebra, derivation, Jordan derivation, higher derivation, generalized higher derivation and so on) and some well-known theorems.In chapter 2, we mainly discuss generalized higher Jordan derivations on trian-gular algebra. We prove that every generalized higher Jordan derivation of triangular algebras is a generalized higher derivation.In chapter 3. we first discuss generalized derivable mappings at zero of triangular algebras. We prove that every generalized derivable map at zero of triangular algebra is a generalized derivation. Subsequently, we discuss Jordan derivable mappings at unit on Triangular algebra and we prove that every Jordan derivable mappings at the unit on triangular algebra is a derivation.