Context
I was reading article [1], where they write Maxwell's equations into the eigenproblem, using Dirac notation, $$ \hat A_k|H_k\rangle = (\omega/c)^2|H_k\rangle, $$ where the field $|H(x,t)\rangle:\mathbb{R}^3\times[0,\infty)\rightarrow\mathbb{R}^3$ has been replaced by the periodic field $|H_k(x)\rangle:\Omega\rightarrow\mathbb{R}^3$, using Bloch's theorem, through $$ |H(x,t)\rangle = e^{i(k\cdot x-\omega t)}|H_k(x)\rangle. $$ Here, $k\in\mathbb{R}^3$ is the so-called Bloch wavevector and $|H_k(x)\rangle$ is a periodic function, "completely defined by its values in the unit cell" $\Omega\subset\mathbb{R}^3$. Furthermore, $|H\rangle$ and $|H_k\rangle$ are divergenceless, i.e., $\nabla\cdot|H\rangle=\nabla\cdot|H_k\rangle=0$ and the operator $\hat A_k$ is defined as $$ \hat A_k := \left(\nabla+ik\right)\times\frac{1}{\varepsilon}\left(\nabla+ik\right)\times, $$ acting on the space that $|H_k\rangle$ lives in. Also, $\varepsilon(x)$ is a function of space. This is a common mathematical context when dealing with electromagnetic fields in periodic media, such as (photonic) crystals.
Question
Anyway, physics aside, the article states that "$\hat A_k$ is the positive semi-definite Hermitian operator", Hermitian meaning self-adjoint. I tried to prove the self-adjoint part (in $L^2(\mathbb{R}^3)$), but I have no intuition how this can be done. Can anyone (loosely) proof the self-adjoint property of $\hat A_k$, assuming any necessary smoothness of functions? If not, any insight on why $\hat A_k$ might be self-adjoint is also welcome!
What I know
So far, I have been able to proof that $i\partial_x$ and $\partial_x^2$, in 1D, are self-adjoint, assuming functions vanish as $x\rightarrow\infty$. To do this, I have used integration by parts to transfer the differential operators from the first to the second function in the inner product. Can something similar be done in $\mathbb{R}^3$?
References
[1] Steven G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis", Optics Express, Vol. 8, No. 3 (2001)