Abstract
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 \(A\in {\mathcal {A}}(X)\).
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ur Rehman, N., Širovnik, N. & Bano, T. On Certain Functional Equations on Standard Operator Algebras. Mediterr. J. Math. 14, 12 (2017). https://doi.org/10.1007/s00009-016-0823-4
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DOI: https://doi.org/10.1007/s00009-016-0823-4