On Certain Functional Equations on Standard Operator Algebras

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• 作者：Nadeem ur Rehman ; Nejc Širovnik ; Tarannum Bano
• 关键词：Prime ring ; standard operator algebra ; derivation ; Jordan derivation ; functional identity
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：14
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1660-5454
• 卷排序：14

文摘

In this paper, functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let \({\mathcal {L}}(X)\) be the algebra of all bounded linear operators of X into itself and let \({\mathcal {A}}(X)\subset {\mathcal {L}}(X)\) be a standard operator algebra. Suppose there exist linear mappings \(D, G:{\mathcal {A}}(X)\rightarrow {\mathcal {L}}(X)\) satisfying the relations \(D(A^{2n+1})=D(A^{2n})A+A^{2n}G(A)\) and \(G(A^{2n+1})=G(A^{2n})A+A^{2n}D(A)\) for all \(A\in {\mathcal {A}}(X)\). Then there exists \(B\in {\mathcal {L}}(X)\) such that \(D(A)=G(A)=[A,B]\) for all \(A\in {\mathcal {A}}(X)\).