A result on functional equations in semiprime rings and standard operator algebras
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Abstract
Let $X$ be a real or complex Banach space, let $L(X)$ be the algebra of all bounded linear operators of $X$ into itself andlet $A(X)\subset L(X)$ be a standard operator algebra. Suppose there exist linear mappings $D,G : A(X) \to L(X)$ satisfying the relations $2D(A^n) = D(A^{n-1})A+A^{n-1}G(A)+ G(A)A^{n-1}+AG(A^{n-1})$ and 2G(A^n) = G(A^{n-1})A+A^{n-1}D(A)+ D(A)A^{n-1}+AD(A^{n-1})$ for all $A\in A(X)$. Then there exists some fixed $B \in L(X)$ such that $D(A) = G(A) = [A;B]$ holds for all $A \in A(X)$.
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Rehman, N., & Bano, T.
(2016).
A result on functional equations in semiprime rings and standard operator algebras.
Acta Mathematica Universitatis Comenianae, 85(1), 21-28.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/105/277
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References
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[10] J. VUKMAN: Identities with derivations and automorphisms on semiprime rings, Internat. J. Math. & Math. Sci. (2005), 1031-1038.
[11] J. VUKMAN: On derivations of algebras with involution, Acta Math. Hungar. 112 (3) (2006), 181-186.
[12] J. VUKMAN: On derivations of standard operator algebras and semisimple H^*-algebras, Studia Sci. Math. Hungar. 44 (2007), 57-63.
[13] J. VUKMAN: Identities related to derivations and centralizers on standard operator algebras, Taiwan. J. Math. Vol. 11 (2007), 255-265.
[14] J. VUKMAN: Some remarks on derivations in semiprime rings and standard operator algebras, Glasnik. Mat. Vol. 46 (2011), 43-48.
[2] M. BREŠAR: Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003-1006.
[3] M. BREŠAR: Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218-228.
[4] P. R. CHERNOFF: Representations, automorphisms and derivations of some Operator Algebras, J. Funct. Anal. 2 (1973), 275-289.
[5] J. CUSACK: Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324.
[6] I. N. HERSTEIN: Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1119.
[7] P. ŠEMRL: Ring derivations on standard operator algebras, J. Funct. Anal. Vol. 112 (1993), 318-324.
[8] N. ŠIROVNIK: On certain functional equations in semiprime rings and standard operator algebras, preprint.
[9] J. VUKMAN: On automorphisms and derivations of operator algebras, Glasnik Mat. Vol. 19 (1984), 135-138.
[10] J. VUKMAN: Identities with derivations and automorphisms on semiprime rings, Internat. J. Math. & Math. Sci. (2005), 1031-1038.
[11] J. VUKMAN: On derivations of algebras with involution, Acta Math. Hungar. 112 (3) (2006), 181-186.
[12] J. VUKMAN: On derivations of standard operator algebras and semisimple H^*-algebras, Studia Sci. Math. Hungar. 44 (2007), 57-63.
[13] J. VUKMAN: Identities related to derivations and centralizers on standard operator algebras, Taiwan. J. Math. Vol. 11 (2007), 255-265.
[14] J. VUKMAN: Some remarks on derivations in semiprime rings and standard operator algebras, Glasnik. Mat. Vol. 46 (2011), 43-48.