On the equivalence of all notions of generalized derivations whose domain is a C^-algebra

Let M be a Banach bimodule over an associative Banach algebra A, and let F: A→ M be a linear mapping. Three main uses of the term generalized derivation are identified in the available literature, namely, () F is a generalized derivation of the first type if there exists a derivation d : A→ M satisfying $F(a b ) = F(a) b + a d(b)$ for all a,b∈ A. () F is a generalized derivation of the second type if there exists an element ξ∈ M** satisfying F(a b ) = F(a) b + a F(b) - a ξ b for all a,b∈ A. () F is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings G,H : A→ M satisfying $F(a b ) = G(a) b + a H(b)$ for all a,b∈ A. There are examples showing that these three definitions are not, in general, equivalent. Despite that the first two notions are well studied when A is a C^*-algebra, it is not known if the three notions are equivalent under these special assumptions. In this note we prove that every generalized derivation of the third type whose domain is a C^*-algebra is automatically continuous. We also prove that every (continuous) generalized derivation of the third type from a C^*-algebra A into a general Banach A-bimodule is a generalized derivation of the first and second type. In particular, the three notions coincide in this case. We also explore the possible notions of generalized Jordan derivations on a C^*-algebra and establish some continuity properties for them.