Question on compactness of generator of unitary group

Question

Let $\mathcal{H}$ be complex Hilbert space. Suppose we have a unitary group $\{U_t\},t\in\mathbb{R}$ and by Stone's Theorem we have a unique infinitesimal generator $A:\mathcal{D}(A)\rightarrow\mathcal{H}$, which is a self-adjoint operator: $$U_t = e^{itA} \quad t\in\mathbb{R}.$$ So, I wonder is this operator $A$ compact or not?
The reason why I ask this is that in P.Walter's Spectral Theorem for Unitary Operators he said "there exists a unique finite Borel measure $\mu_f$", however, in other materials, the term "spectral measure" is used. I noticed that the Borel measure is a spectral measure if there exists an orthogonal basis formed by spectrum of $A$, but then I also see here that the compact self-adjoint operator $A$ can have such result. So, can someone explain me more about the details here why P.Walters theorem is different from others?