What can be expected from the image of $\sigma$-derivations on Banach algebras?

The main purpose of this paper is to obtain some results on the image of $\sigma$-derivations on Banach algebras. One of the main results of this paper is to prove that if $\mathcal{A}$ is a commutative Banach algebra and $d:\mathcal{A} \rightarrow \mathcal{A}$ is a continuous $\sigma$-derivation such that $\sigma$ is a continuous homomorphism, $d \sigma = \sigma d = d$ and $\sigma^{2} = \sigma$, then $d(\mathcal{A}) \subseteq {\rm rad}(\mathcal{A})$, where ${\rm rad}(\mathcal{A})$ denotes the Jacobson radical of $\mathcal{A}$. Moreover, we obtain Sinclair's theorem for $\sigma$-derivations without assuming continuity. Indeed, under certain conditions, we prove that if $d$ is a $\sigma$-derivation on a Banach algebra $\mathcal{A}$, then $d(\mathcal{P}) \subseteq \mathcal{P}$ for every primitive ideal $\mathcal{P}$ of $\mathcal{A}$. Some other related results are also discussed.