= Solution
The <Stone theorem for the circle group> says that a strongly continuous <unitary representation> $U$ of $\mathbb T=\{z:|z|=1\}$ has an orthogonal decomposition
$$
\boxed{H=\bigoplus_{n\in\mathbb Z}H_n,\qquad U_z|_{H_n}=z^nI.}
$$
Equivalently, $U_z=\sum_nz^nP_n$ strongly, with mutually <orthogonal projections> $P_n$ summing strongly to the identity. Define these <projections> using normalized <Haar measure>:
$$
P_n\xi=\int_{\mathbb T}z^{-n}U_z\xi\,dz.
$$
Haar invariance gives $U_wP_n=w^nP_n$. Changing $z$ to $z^{-1}$ in the adjoint integral proves $P_n^*=P_n$. Applying the defining integral to $P_m\xi$ gives $P_nP_m=\delta_{nm}P_m$ by <orthogonality> of the characters $z^k$. Thus $P_n$ are <orthogonal projections>, and their ranges are exactly the indicated character subspaces.
It remains to prove completeness of these subspaces. The <Fejér kernel> has the finite expansion
$$
K_N(e^{it})=\sum_{|n|<N}\left(1-\frac{|n|}{N}\right)e^{-int}
=\frac1N\left|\sum_{j=0}^{N-1}e^{ijt}\right|^2.
$$
It is nonnegative, has normalized integral one, and its integral outside any fixed neighborhood of the identity tends to zero, since the denominator in the usual geometric-sum formula is bounded away from zero there. Strong <continuity> therefore gives
$$
\int K_N(z)U_z\xi\,dz\longrightarrow\xi:
$$
split the integral into that neighborhood, where $\|U_z\xi-\xi\|$ is small, and its complement, where it is bounded by $2\|\xi\|$. The integral on the left is $\sum_{|n|<N}(1-|n|/N)P_n\xi$. Hence the closed span of all $H_n$ is $H$. <Orthogonality> then gives the unweighted strong sum and the stated representation formula.
Conversely, any orthogonal decomposition indexed by the integers gives this representation; <continuity> follows first on finite sums and then on all <vectors> by the uniform unitary <norm> bound. With $z=e^{it}$ its generator is
$$
A\xi=\sum_nnP_n\xi,\qquad
D(A)=\left\{\xi:\sum_nn^2\|P_n\xi\|^2<\infty\right\}.
$$
This diagonal operator is <self-adjoint> and $U_{e^{it}}=e^{itA}$. Thus the circle theorem is the integer-spectrum version of the one-parameter theorem.
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