Approximation of solutions of the heat equation of Lebesgue class $L^2$ by more regular solutions

Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and let $\Omega, \omega $ be bounded domains with smooth boundaries in ${\mathbb R}^n$, $n \geq 1$ such that $\omega \subset \Omega$. We prove that the space $H^{2s,s} _{\mathcal H} (\Omega \times (T_1,T_2))$ of solutions of the heat operator ${\mathcal H} = \frac{\partial}{\partial t} - \sum_{j=1}^n \frac{\partial^2}{\partial x^2_j}$ in the cylinder domain $\Omega \times (T_1,T_2)$ belonging to anisotropic Sobolev space $H^{2s,s} (\Omega \times (T_1,T_2))$ is everywhere dense in the space $L^{2} _{\mathcal H}(\omega \times (T_1,T_2))$, consisting of solutions in the domain $\omega \times (T_1,T_2)$ of the Lebesgue class $L^{2} (\omega \times (T_1,T_2))$, if and only if the complement $\Omega \setminus \omega$ has no compact components in $\Omega$. As an important corollary we obtain the theorem on the existence of a basis with the double orthogonality property for the pair of the Hilbert spaces $H^{2s,s} _{\mathcal H} (\Omega \times (T_1,T_2))$ and $L^{2} _{\mathcal H}(\omega \times (T_1,T_2))$.