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

Let X be a locally compact Hausdorff topological space, let V = ( v n , k ) ∈ N $\mathcal {V}=(v_{n,k})_{n,k\in {\mathbb {N}}}$ system of positive continuous functions on and φ self-map X. The composition operators C : f ↦ ∘ $C_\varphi f\mapsto \ f\,\circ\, \varphi$ the weighted function (LF)-spaces {V}C(X)$ 0 {V}_0C(X)$ resp.) (PLB)-spaces A {A}C(X)$ {A}_0C(X)$ are studied. We characterize when operator $C_\varphi$ acts continuously such spaces in terms {V}$ map φ, as well we determine conditions which correspond to various basic properties like boundedness, compactness, weak compactness. Our approach requires study continuity, (weak) compactness linear between (PLB)-spaces.