Spectral measures and automatic continuity (Q1815268)

Let \(\Sigma\) be a \(\sigma\)-algebra of subset of a set \(S\), \(X\) be a locally convex Hausdorff space and \(L(X)\) the space of all continuous linear operators from \(X\) into itself equipped with the strong operator topology. Let \(P:\Sigma\to L(X)\) be countably additive such that \(P(E\cap F)= P(E)P(F)\) holds for every \(E,F\in\Sigma\) and \(P(S)=I\), the identity operator (i.e., \(P\) is a spectral measure). For each \(x\in X\) let \(P_x\) be the vector measure on \(\Sigma\) defined by \(P_x(E)= P(E)x\). If \(f\) is \(P\)-integrable, then \(f\) is \(P_x\)-integrable for every \(x\in X\) and the integration map \(\int f dP\) is continuous on \(X\) and \(\int f dP\) satisfies \((\int f dP)x=\int f dP_x\) for every \(x\in X\). The authors address the converse. Namely, if \(f\) is \(P_x\) integrable for every \(x\in X\) when is the integration map \(P_f: X\to X\), \(P_f(x)=\int f dP_x\), continuous. The authors establish the automatic continuity of the map \(P_f\) under 3 separate hypotheses; for example, they show this is the case if \(X\) is barrelled. These results improve results of Dodds and de Pagter. Examples are given relevant to the various hypotheses in the theorems.