If P : Σ →ℒ(X) is a closed
spectral measure in the quasicomplete locally convex space X and if T is a densely
defined linear operator in X with domain invariant under each operator of the form
∫Ωf dP, with f a complex bounded Σ-measurable function then T is closable and
there exists a complex Σ-measurable function f such that the closure of T is
the spectral integral ∫Ωf dP if and only if T leaves invariant each closed
subspace of X which is invariant under the range of the spectral measure
P.