Operators $A_i, i=1,2,\ldots,5$ are considered on Hilbert spaces $\mathcal{H}_i$. Operator $A_i$ is essentially self-adjoint on $D^{ess}(A_i) \subset \mathcal{H}_i$.
Consider operator \begin{equation} L = A_1 \otimes A_2 \otimes I \otimes I \otimes I + I \otimes M \otimes A_3 \otimes A_4 \otimes A_5, \end{equation} where $I_k$ is an identity operator on $\mathcal{H}_k$ and $M$ is a multiplication operator on a real-valued bounded function such that $M: \mathcal{H}_2 \rightarrow \mathcal{H}_2$.
Is operator $L$ essentially self-adjoint on $D^{ess}(A_1) \otimes D^{ess}(A_2) \otimes \ldots \otimes D^{ess}(A_5)$?
