Operator gradient of divergencie in subspaces of $\mathbf{L}_{2}(G)$ space

The author studies the structure of space $\mathbf {L} _ {2} (G)$ of vector-valued functions that are square integrable in a bounded connected domain $G$ of the three-dimensional space with a smooth boundary and the role of gradient divergence operators and the rotor in the construction of bases in subspaces ${\mathcal {{A}}}$ and ${\mathcal {{B}}}$. The self-adjointness of the extension $\mathcal {N} _d$ of operator $\nabla \mathrm {div}$ to the subspace $\mathcal {A} _ {\gamma} \subset {\mathcal {{A}}}$ and the basicity system of its own functions. Written explicit formulas for solving the spectral problem in a ball and the conditions for the decomposition vector-functions in a Fourier series in eigenfunctions gradient of divergence. The solvability of the boundary tasks: $\nabla \mathrm {div} \, \mathbf {u} + \lambda \, \mathbf {u} = \mathbf {f}$ in $G$, $(\mathbf {n} \cdot \mathbf {u}) | _ {\Gamma} = g$ in Sobolev spaces $\mathbf {H} ^ {s} (G)$ of order $s \geq 0$ and in subspaces. In passing, similar results for the operator of the rotor and its symmetric extension $S$ to $\mathcal {B}$.