On composition operators between weighted (LF)- and (PLB)-spaces of continuous functions

Let X be a locally compact Hausdorff topological space, let $\mathcal {V}=(v_{n,k})_{n,k\in {\mathbb {N}}}$ be a system of positive continuous functions on X and let φ be a continuous self-map on X. The composition operators $C_\varphi : f\mapsto \ f\,\circ\, \varphi$ on the weighted function (LF)-spaces $\mathcal {V}C(X)$ ($\mathcal {V}_0C(X)$, resp.) and on the weighted function (PLB)-spaces $\mathcal {A}C(X)$ ($\mathcal {A}_0C(X)$, resp.) are studied. We characterize when the operator $C_\varphi$ acts continuously on such spaces in terms of the system $\mathcal {V}$ and the map φ, as well as we determine conditions on $\mathcal {V}$ and φ which correspond to various basic properties of the composition operator $C_\varphi$, like boundedness, compactness, and weak compactness. Our approach requires a study of the continuity, boundedness, (weak) compactness of the linear operators between (LF)-spaces and (PLB)-spaces.