Solving the Sylvester equation in Banach modules

For given unital complex Banach algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, let $\mathfrak{M}$ be a Banach module acting between them. Let $a\in \mathcal{A}_1$, $b\in\mathcal{A}_2$, and $c\in\mathfrak{M}$ be provided such that $\sigma_{\mathcal{A}_1}(a)\cap\sigma_{\mathcal{A}_2}(b) \neq\emptyset$. In this paper we completely characterize the consistency of the Sylvester equation $$ax-xb=c.$$ Precisely, we establish verifiable sufficient and necessary solvability conditions, and we provide some formulas for particular solutions $x\in\mathfrak{M}$ when the equation is solvable. Moreover, we characterize the uniqueness of the solutions.