Stone's theorem I have some basic doubts about Stone's theorem.
1) Can we apply Stone's theorem to conclude that given any Unitary operator U, we can find a self adjoint operator A such that  U = exp(i A). That is, is any unitary operator part of a one parameter group?
2) Is there some version of Stone's theorem for real Hilbert spaces like in finite dimensions?
 A: Stone's theorem works in both senses. Given a one-parameter unitary group of operators $U(t)$ on a Hilbert space, there is one self-adjoint operator $H$, called its generator, such that $i\partial_t U(t)=HU(t)$. Conversely, given a self-adjoint operator we can define a unitary group of operators generated by it.
Stone's theorem obviously works also on real Hilbert spaces.
A: As mentioned in the previous post, you won't have uniqueness of the unitary group. But I think you can find one using the spectral theorem for $U$. To find $U$, write
$$
            U = \int_{T} \lambda dE(\lambda),
$$
where $T$ is the unit circle in the complex plane, and $E$ is the Borel spectral measure for $U$. Define
$$
                      U(t) = \int_{T} e^{it\arg(\lambda)} dE(\lambda),
$$
where, for example, you take $\arg(\lambda) \in [0,2\pi)$ such that $e^{i\arg(\lambda)}=\lambda$ on $T$. The generator $A$ of this semigroup is the bounded selfadjoint operator $A=\int_{T}\arg(\lambda)dE(\lambda)$. Notice that $\sigma(A)\subseteq [0,2\pi]$ and $2\pi \notin\sigma_{p}(A)$, which may make such $A$ unique. (?)
