GENERALIZED ¾-DERIVATION

Let A be a Banach algebra and M be a Banach A-bimodule. We say that a linear mapping ±:A!M is a generalized ¾-derivation whenever there exists a ¾-derivation d:A!M such that ±(ab) = ±(a)¾(b) + ¾(a)d(b) for all a,b2 A. Giving some facts concerning generalized ¾- derivations, we prove that, if A is unital and ±:A!A is a generalized ¾- derivation, and if there exists an element a 2 A such that d(a) is invertible, then ± is continuous if and only if d is continuous. Finally, a formula will be introduced concerning ±n(ab) for all n 2 N and a; b 2 A, where ± : A ! A is a generalized ¾-derivation