Weakly compact weighted composition operators on certain subspaces of \(C(X,E)\) (Q1185158)

Let \(X\) be a compact Hausdorff space and \(E\) a complex Banach space. Let \(A\) be a function algebra, i.e. a uniformly closed subalgebra of \(C(X)\) which separates points of \(X\) and contains the constants. Denote by \(A(X, E)\) the set of all continuous functions \(f: X\to E\) such that \(e^*\circ f\in A\) for every \(e^*\in E^*\). A bounded linear operator \(T: A(X, E)\to A(X, E)\) is called a weighted composition operator if it has the form \(Tf(x)= w(x) f(\varphi(x))\) \((x\in X,\;f\in A(X, E))\), where \(\varphi: X\to X\) and \(w: X\to {\mathcal L}(E)\) are some maps and \({\mathcal L}(E)\) denotes the algebra of all bounded linear operators on \(E\). The paper studies compact and weakly compact weighted composition operators on \(A(X, E)\).