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Spectral measures and automatic continuity. (English) Zbl 0890.46030
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.

46G10 Vector-valued measures and integration
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
46A08 Barrelled spaces, bornological spaces
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