Stone theorem for the circle group ($U_z={\sum }_{n\in \mathbb{Z}}{z}^n{P}_n$)

A strongly continuous unitary representation of the circle group splits as an orthogonal sum of character subspaces. The projections are $P_n={\int }_{\mathbb{T}}{z}^{-n}{U}_{z}dz$. Character orthogonality makes them mutually orthogonal, and Fejér sum shows their ranges span the represented Hilbert space. The self-adjoint generator has integer eigenvalues, with domain given by ${\sum }_n{n}^2\Vert P_n\xi {\Vert }^2<\infty$.