When is an operator the integral of a given spectral measure?

AbstractFor a closed densely defined operator T on a complex Hilbert space H and a spectral measure E for H of countable multiplicity q defined on a σ-algebra B over an arbitrary space Λ we give three conceptually differing but equivalent answers to the question asked in the title of the paper (Theorem 1.5). We then study the simplifications which accrue when T is continuous or when q = 1 (Sect. 4). With the aid of these results we obtain necessary and sufficient conditions for T to be the integral of the spectral measure of a given group of unitary operators parametrized over a locally compact abelian group Γ (Sect. 5). Applying this result to the Hilbert space H of functions which are L2 with respect to Haar measure for Γ, we derive a generalization of Bochner's theorem on multiplication operators (Sect. 6). Some results on the multiplicity of indicator spectral measures over Γ are also obtained. When Γ = R we easily deduce the classical theorem about the commutant of the associated self-adjoint operator (Sect. 7)