Consider some unitary which evolves a system according to
$$\hat{U}(\varphi) = \exp\left[-i\hat{G}(\varphi)\right],$$
where $\hat{G}$ is the Hamiltonian of the system. Then by spectral decomposition (i.e. eigenvalue/eigenvector decomposition),
$$\hat{G}(\varphi) = \sum_{j=1}^{n_g} \sum_{k=1}^{d_j} E_j(\varphi)\vert E_j^{(k)}(\varphi)\rangle\langle E_j^{(k)}(\varphi)\vert. $$
Then we can derive the generator for the system:
\begin{align} \begin{split} \mathscr{\hat{G}}(\varphi) & = \sum_{k = 1}^{n_g}\frac{\partial E_k}{\partial \varphi}\hat{P}_k + 2\sum_{k\neq l}\sum_{i=1}^{d_k}\sum_{j=1}^{d_l} \exp\left[-i (E_k - E_l)/2\right] \\ & \times \sin\left[\frac{E_k - E_l}{2}\right] \langle E_l^{(j)}\vert\partial_\varphi E_k^{(i)}\rangle\vert E_k^{(i)}\rangle\langle E_l^{(j)}\vert, \end{split} \end{align}
where $\smash{\hat{P}_k = \sum_j\vert E_k^{(j)}\rangle\langle E_k^{(j)}\vert}$ is the projector onto the eigenspace of the operator $\hat{G}(\varphi)$.
Any meaningful observable requires this operator to be self-adjoint (or Hermitian for non-infinite Hilbert spaces). From inspection, it is clear that it is not self-adjoint. Under what constraints to $\varphi$ can it be considered self-adjoint?
I have had a look at this Wiki page on Self-adjoint operators, but I have been unsuccessful. I believe that if $\mathscr{\hat{G}}(\varphi)$ demonstrates some periodicity with $\varphi$ then by choosing a suitable interval, self-adjointness may follow.
Can anyone demonstrate this intuition or alternatively direct me along a more fruitful path? Thanks.