Multiplication Operator Semigroups on Banach lattice valued continuous function spaces

We introduce and characterize, on the Banach lattice valued continuous function space, multiplication operators generating strongly continuous multiplication operator semigroups. Our characterization is the generalization of known results for the scalar-valued continuous functions $C_0(\mathrm{\Omega })$ vanishing at infinity, on a locally compact (Hausdorff) space $\mathrm{\Omega }$, to Banach lattice $C_0(\mathrm{\Omega },E)$ of continuous Banach lattice $E$-valued functions vanishing at infinity.