The curl operator in the L2(G) space

Author studies properties of the curl and gradient of divergence operators in the $\mathbf{L}_2(G)$ space, spectral decompositions, and boundary value problems for any bounded domain $G$ with smooth boundary $\Gamma$.
It turns out that the space $\mathbf{L}_2(G)$ has orthogonal subspaces $\mathbf{V}^0(G)$ and $\mathscr A_\gamma(G)$ such that the curl and gradient of divergence operators admit self-adjoint extensions.
Therefor, each of these operators has a complete system of eigenfunctions corresponding to non zero eigenvalues.
These results supplement Weil’s well known theorem on a decomposition of $\mathbf{L}_2(G)$ on orthogonal subspaces $\mathscr A_\gamma(G)$, $\mathbf{V}^0(G)$ and $\mathscr B_H(G)$ of finite dimension. It show that the space $\mathbf{L}_2(G)$ has a bases consisting of eigenfunctions of the curl and gradient of divergence operators.
We investigate also the solvability of the boundary value problem $\mathbf{rotu}+\lambda \mathbf{u}=\mathbf{f}$ in $G$, $\mathbf{n}\cdot\mathbf{u}|_\Gamma = g$, for $\lambda\neq 0$ and (by Fourier method) in a ball $G=B$ for all $\lambda$.
Key words: the curl and gradient of divergence operators, $\mathbf{L}_2(G)$ space, spectral decompositions, boundary value problem, the bounded domain $G$, smooth boundary $\Gamma$.