Stone’s theorem for a group of unitary operators over a Hilbert space

Abstract

The spectral representation for a group of unitary operators acting on a Hilbert space where the parameter set is a separable real Hilbert space is obtained. The usual spectral representation of such a group of unitary operators is when the parameter set is a locally compact abelian group (Stone’s theorem). The main result used in the proof is the Bochner theorem on the representation of positive definite functions on a real Hilbert space.