Spectral measures and the Bade reflexivity theorem (Q1066055)

Let X be a quasicomplete locally convex topological vector space and let \({\mathcal L}(X)\) be the space of continuous linear operators on X, assumed sequentially complete for the strong operator topology. The Boolean algebra \({\mathcal B}\) of projections in X is called strongly equicontinuous if \({\mathcal B}\) is an equicontinuous subset of X and if each disjoint sequence in \({\mathcal B}\) is pointwise convergent to zero. Methods of integration with respect to spectral measures are used to show that the closed algebra generated by \({\mathcal B}\) in \({\mathcal L}(X)\) consists precisely of those continuous linear operators on X which leave invariant each closed \({\mathcal B}\)-invariant subspace of X.