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 ${\sigma }_{{\mathcal{A}}_1}(a)\cap {\sigma }_{{\mathcal{A}}_2}(b)\mathrm{\ne }\mathrm{\varnothing }$. In this paper we completely characterize the consistency of the Sylvester equation $$ax-xb=c\mathrm{.}$$ 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.