Criteria for closedness of spectral measures and completeness of Boolean algebras of projections
Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of some non-empty set. Let \(Y\) be a locally convex Hausdorff space with dual \(Y'\). A \(\sigma\)-additive measure \(m: \Sigma \to Y\) is called closed if the set of images of \(\Sigma\) under \(m\) is complete with respect to a family of metric induced by the total variation measures \(| \langle m(\cdot)y, y'\rangle| \) with \(y \in Y\), \(y' \in Y'\). It is known that \(m\) is closed if it is countably determined in the sense that there is a sequence \(y_n'\), \(n = 1,2, \ldots\), such that \(\langle m(E),y_n'\rangle = 0\) for all \(n\) implies \(m(E) = 0\). In the present paper the authors confine their attention to the case where \(Y\) is the space \(L_s(X)\) of all continuous linear maps on a locally convex Hausdorff space \(X\), endowed with the strong operator topology, and \(m\) is a spectral measure, that is, a multiplicative measure \(P: \Sigma \to L_s(X)\). They investigate various sufficient conditions for closedness of a spectral measure \(P\). Here are some of such sufficient conditions: (a) there is a numerical measure \(\lambda\) on \(\Sigma\) such that every measure \(\langle P(\cdot)x,x'\rangle\) is absolutely continuous with respect to \(\lambda\), (b) \(P\) has a separating vector \(x\) such that the vector measure \(P(\cdot)x: \Sigma \to X\) is countably determined, (c) \(P\) has a countable separating set of vectors and there is a metrizable locally convex topology on \(X\) which is weaker than the given topology. Here a sequence \(x_n\), \(n = 1,2, \ldots\), is separating for \(P\) if \(P(E)x_n = 0\) for all \(n\) implies \(P(E) = 0\).
scientific article; zbMATH DE number 1278867
scientific article