Identities related to generalized derivations and Jordan (∗,⋆)-derivations

Abstract

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) = −x2G(x−1) for all
x ∈ Inv(A). Then both F and G are generalized (left) derivations on A. Moreover, we define a (∗,⋆)-ring
and a Jordan (∗,⋆)-derivation and present a characterization of Jordan (∗,⋆)-derivations as follows. Let
R be an n!-torsion free (∗, ◦)-ring, n ≥ 1 be an integer and let d : R → R be an additive mapping satisfying
d(an) =
Σnj
=1 a⋆n−jd(a)a∗ j−1 for all a ∈ R. Then d is a Jordan (∗,⋆)-derivation on R. Some other functional
identities are also investigated.