Invertible weighted composition operators on weighted function spaces
If \(X\) is a completely regular Hausdorff space and \(E\) a locally convex Hausdorff topological vector space, let \(C(X, E)\) denote the space of all continuous functions from \(X\) into \(E\). Also let \(L(X)\) be a topological vector subspace of the Cartesian product \(\prod_{x\in X} F_x\) where \((X, (F_x )_{x\in X})\) is a vector filtration over \(X\). For suitable systems \(V\) of weights on \(X\), the authors define weighted spaces of continuous functions \(CV_b (X, E)\) and \(CV_0 (X, E)\) (both are contained in \(C(X, E)\)), and weighted spaces of cross-sections over \(X\), \(LV_b (X)\), \(LV_0 (X)\) (both are contained in \(L(X)\)). These are locally convex spaces under suitable seminorms. For \(\varphi: X\to X\) and \(\pi: X\to E\) continuous mappings define the weighted composition operator \(W_{\pi, \varphi} (f)= \pi(f \circ \varphi)\). The authors characterize those \(W_{\pi, \varphi}\) that are invertible as operators on \(CV_0 (X, E)\) or on \(LV_0 (X)\).
locally convex Hausdorff topological vector space
scientific article; zbMATH DE number 721632
scientific article