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 $\mathrm{\nabla }\mathrm{d}\mathrm{i}\mathrm{v}$ 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 $inSobolevspaces$ \mathbf {H} ^ {s} (G) $oforder$ s \geq 0 $ and in subspaces. In passing, similar results for the operator of the rotor and its symmetric extension $S$ to $\mathcal{B}$.