The Higher Derivable Mapping At Some Fixed Points

In recent years, the questions of revealing the relationships of derivations andstudying all-derivable points have brought many mathematicians’attention andinterest, and they have obtained a lot of results. With the development of derivations,there also has been considerable interest in studying higher derivations or higherJordan derivations. Zhang Jianhua[1]showed that every Jordan derivable mapping inthe triangular algebra is an inner derivation. Qi and Hou[2]proved that an additive mapL is an additive (generalized) Lie derivation if and only if it is the sum of an additive(generalized) derivation and an additive map from the algebra into its center vanishingall commutators. Zhu Jun[3]proved that the unit operator I is an all-derivable pointof nest algebras for the strongly operator topology. Lu Fangyan[4]showed that inBanach space every idempotent is an all-derivable point. Jin Wu[5]proved that theunit operator I is an Jordan all-derivable point of B(H) . Xiao Zhankui and WeiFen[6]proved that every higher Jordan derivation in the triangular algebra is a higherderivation.Lately, Hou Jinchuan[7]have proved that some idempotents are all-derivablepoints on rings. Zhu Jun and Xiong Changping have proved that (1)G∈TM2is anall-derivable point ofTM 2if and only if G≠0whereTM 2is the algebra of alln×nupper triangular matrices[8]. (2)G∈Unis an all-derivable point inU nif andonly if G≠0whereU nis the algebra of all n×nmatrices[9].In recent yearsthere has been considerable interest in studying the questions of generalizedderivations and higher derivations.It has four chapters in this article. In the first chapter, it presents the relatedsymbols, definitions and the background of the research. Motivated by HouJinchuan’s paper, the second chapter gives some all-derivable points in triangularalgebras. the second chapter gives some all-derivable points in triangular algebras. Inthe third chapter, we get some Jordan all-derivable points in triangular algebras byusing the way of induction. The paper has proved: (1)0 is a all derivable point；is a Jordan all-derivable point. In the forth charpter, wecharacterize the derivation on the Banach algebras.