Closedness of sums of unbounded operators acting on different variables in the spaces of square-integrable functions of several variables

Let $\Omega'$ and $\Omega''$ be some Lebesgue measurable sets in the metric spaces $\mathbf{R}^m$ and $\mathbf{R}^n$, respectively, and $\mathbf{A}$ be a linear operator in the space $L_2(\Omega')$. We define an operator $\tilde{\mathbf{A}}$ in the space $L_2(\Omega'\times \Omega'')$ on the basis of equalities
$$ (\tilde{\mathbf{A}}\mathbf{f})(x'')=\mathbf{A}\mathbf{f}(x'')\quad (\mathbf{f}\in D(\tilde{\mathbf{A}}),\ x''\in\Omega''), $$
where $D(\tilde{\mathbf{A}})$ is a domain of operator $\tilde{\mathbf{A}}$. These equations mean that an element $\mathbf{f}\in L_2(\Omega'\times \Omega'')$ represented by a function $\mathbf{f}(x'')$ with values in $D(\mathbf{A})$ belongs to the set $D(\tilde{\mathbf{A}})$ if there exists an element $\mathbf{g}\in L_2(\Omega'\times \Omega'')$ represented by the function $\mathbf{g}(x'')$ such that the pointwise equalities $\mathbf{g}(x'')=\mathbf{A}\mathbf{f}(x'')$ are satisfied almost everywhere in the Lebesgue measure on the set $\Omega''$. Then, $\tilde{\mathbf{A}}\mathbf{f}=\mathbf{g}$. Similarly, using a linear operator $\mathbf{B}$ acting in the space $L_2(\Omega'')$, we define an operator $\tilde{\mathbf{B}}$ in the space $L_2(\Omega'\times \Omega'')$. It is proved that the sum of operators $\tilde{\mathbf{A}}+\tilde{\mathbf{B}}$ defined on the set $D(\tilde{\mathbf{A}})\cap D(\tilde{\mathbf{B}})$ is closed if the operators $\mathbf{A}$ and $\mathbf{B}$ are generators of some $C_0$-semigroups of contractions; here, the operator $\mathbf{B}$ is selfadjoint and has a purely point spectrum. For example, if the operator $\mathbf{A}_t$, $(\mathbf{A}_t\mathbf{f})(t)=f'(t)$ is defined on absolutely continuous functions $f(t)\in L_2(I_T)$ ($I_T\equiv [0, T]$) such that $f'(t)\in L_2(I_T)$ and $f(0)=0$, as well as on equivalent functions and operator $\mathbf{B}_y$, $(\mathbf{B}_y\mathbf{f})(y)=-f''(y)$, is defined on absolutely continuously differentiable functions $f(y)\in L_2(I_Y)$ ($I_Y\equiv [0, Y]$) such that $f''(y)\in L_2$ and $f'(0)-\lambda_0 f(0)=0$, $f'(0)+\lambda_Yf(0)=0$ ($0\leqslant \lambda_0$, $\lambda_Y\leqslant\infty$), as well as on equivalent functions, the sum of differential operators $\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y$ is closed. The closure of the operator $\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y$ is used as a coefficient in operator-differential equations in the formulation of problems of multidimensional non-stationary heat conduction. We have studied smoothness of functions included in the domains of powers of operators $\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y$. It is proved that if $f(y, t)\in D\left((\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y)^n\right)$ ($n\geqslant 2$), then, almost everywhere on the set $I_Y\times I_T$, there exist derivatives $\partial_t^{l-1}\partial_y^{2(n-l)-1}f$ ($l=\overline{1, n-1}$) equivalent to functions absolutely continuous on $I_Y\times I_T$.