Identities Related to Generalized Derivations and Jordan (*,*)-Derivations

The main purpose of this research is to characterize generalized (left) derivations and Jordan (*,*)-derivations on Banach algebras and rings using some functional identities. Let A be a unital semiprime Banach algebra and let F,G : A -> A be linear mappings satisfying F(x) =-x(2)G(x(-1)) for all x is an element of Inv(A), where Inv(A) denotes the set of all invertible elements of A. Then both F and G are generalized derivations on A. Another result in this regard is as follows. Let A be a unital semiprime algebra and let n > 1 be an integer. Let f, g : A -> A be linear mappings satisfying f (a(n)) = na(n-1)g(a) = ng(a)a(n-1) for all a is an element of A. If g(e) is an element of Z(A), then f and g are generalized derivations associated with the same derivation on A. In particular, if A is a unital semisimple Banach algebra, then both f and 1 are continuous linear mappings. Moreover, we define a (*,*)-ring and a Jordan (*,*)-derivation. A characterization of Jordan (*,*)-derivations is presented as follows. Let R be an n!-torsion free (*,*)-ring, let n > 1 be an integer and let d : R -> R be an additive mapping satisfying d(a(n)) = Sigma(n)(j =1) a*(n-j)d(a)a*( j-1) for all a is an element of R. Then d is a Jordan (*,*)-derivation on R. Some other functional identities are also investigated.