Let $mathcal{A}$ be a Banach algebra and $mathcal{M}$ be a Banach $mathcal{A}$-bimodule. We say that a linear mapping $delta:mathcal{A} rightarrow mathcal{M}$ is a generalized $sigma$-derivation whenever there exists a $sigma$-derivation $d:mathcal{A} rightarrow mathcal{M}$ such 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 $mathcal{A}$ is unital and if $delta:mathcal{A} rightarrow mathcal{A}$ is a generalized $sigma$-derivation and there exists an element $a in mathcal{A}$ such that emph{d(a)} is invertible, then $delta$ is continuous if and only if emph{d} is continuous. We also show that if $mathcal{M}$ is unital, has no zero divisor and $delta:mathcal{A} rightarrow mathcal{M}$ is a generalized $sigma$-derivation such that $d(textbf{1}) neq 0$, then $ker(delta)$ is a bi-ideal of $mathcal{A}$ and $ker(delta) = ker(sigma) = ker(d)$, where textbf{1} denotes the unit element of $mathcal{A}$.
Hosseini, A., Hassani, M., & Niknam, A. (2011). Generalized sigma-derivation on Banach algebras. Bulletin of the Iranian Mathematical Society, 37(No. 4), 81-94.
MLA
A. Hosseini; M. Hassani; A. Niknam. "Generalized sigma-derivation on Banach algebras". Bulletin of the Iranian Mathematical Society, 37, No. 4, 2011, 81-94.
HARVARD
Hosseini, A., Hassani, M., Niknam, A. (2011). 'Generalized sigma-derivation on Banach algebras', Bulletin of the Iranian Mathematical Society, 37(No. 4), pp. 81-94.
VANCOUVER
Hosseini, A., Hassani, M., Niknam, A. Generalized sigma-derivation on Banach algebras. Bulletin of the Iranian Mathematical Society, 2011; 37(No. 4): 81-94.