counter example of sum of closable operators

Question

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a counter example where equality does not hold.

Answer 1

Let $X = L^2[-\pi,\pi]$, and let $A=\frac{d^2}{dx^2}$ on the domain consisting of all polynomials. Let $B=\frac{d^2}{dx^2}$ on the domain $\mathcal{D}(B)$ of all linear combinations of $\{ \sin(t),\sin(2t),\ldots \}$. Both $A$ and $B$ are densely-defined, and they're both closable. However $\mathcal{D}(A)\cap\mathcal{D}(B)=\{0\}$, which forces $\mathcal{D}(A+B)=\{0\}$. However, $\overline{B}\subset\overline{A}$, which gives $\overline{A}+\overline{B}=2\overline{B}$. So, $$ \overline{A+B} \subsetneq \overline{A}+\overline{B}=2\overline{B}. $$