Compact weighted composition operators on certain subspaces of \(C(X,E)\) (Q1176125)

Let be \(A(X,E)=\{f\in C(X,E):\;e^*\circ f\in A\) for all \(e^*\in E^*\}\), where \(X\) is a compact Hausdorff space, \(E\) a complex Banach space, \(C(X,E)\) the Banach space of all continuous \(E\)-valued functions on \(X\) (\(\| f\|=\sup\{\| f(x)\|_ E:\;x\in X\)) and \(E^*\) the dual space of \(E\). Let \(T=wC_ \varphi\) be a weighted composition operator on \(A(X,E)\), meaning ``a bounded linear operator \(T: A(X,E)\to A(X,E)\) which has the form \(Tf(x)=w(x)f(\varphi(x))\), \(x\in X\), \(f\in A(X,E)\), for some selfmap \(\varphi\) of \(X\) and some map \(w\) from \(X\) into \(B(E)\), the space of bounded linear operators on \(E\)''. The author's main theorem shows that, if \(wC_ \varphi\) is compact, three consequences are true and also that the consequences, with an additional property, assure the compactness of \(wC_ \varphi\). One of the two corollaries affirms that ``if \(E\) is infinite dimensional, then there is no compact composition operator on \(A(X,E)\)''.