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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.

##### MSC:
 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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