Generalized σ-derivation on Banach algebras

Let A be a Banach algebra and M be a Banach A-bimodule. We say that a linear mapping delta:mathcal{A} rightarrow M is a generalized sigma-derivation whenever there exists a sigma-derivation d:A rightarrow Msuch that delta(ab) = delta(a) sigma(b) + sigma(a)d(b), for all a,b in mathcal A. Giving some facts concerning generalized sigma-derivations, we prove that if A is unital and if delta:A rightarrow A is a generalized sigma-derivation and there exists an element a in A such that emph{d(a)} is invertible, then $delta$ is continuous if and only if emph{d} is continuous. We also show that if M is unital, has no zero divisor and delta:A rightarrow M is a generalized sigma-derivation such that d(1) neq 0, then ker(delta) is a bi-ideal of A and ker(delta) = ker(sigma) = ker(delta), where 1 denotes the unit element of {A}.