Extensions of certain compact operators on vector-valued continuous functions

Abstract:

For any compact Hausdorff spaces $X,Y$ with $\varphi :\;X \to Y$ a continuous onto mapping, $E,F$, Hausdorff locally convex spaces with $F$ complete, $C(X,E)\;(C(Y,E))$ all $E$-valued continuous functions on $X(Y)$, and $L:C(Y,E) \to F$ a $\mathcal {T}$-compact continuous operator $(\sigma (F,F’) \leq \mathcal {T} \leq \tau (F,F’))$, it is proved there exists a $\mathcal {T}$-compact continuous operator ${L_0}:C(X,E) \to F$ such that ${L_0}(f \circ \varphi ) = L(f)$ for every $f \in C(Y,E)$.