I found this question on another forum, and after processing it a bit, I didn't find a good answer. The question is:
Is the sum of two closed operators closable? If not, give an example of two closed operators such that their sum is not closable.
I found this question on another forum, and after processing it a bit, I didn't find a good answer. The question is:



I might change your question a little bit: Let $X=L^2[0,1]$, $Af:=f''$ with $D(A)=H^2[0,1]$ and let $Bf=f'(0)\cdot\mathbb{1}$, with $D(B)=H^2[0,1]$. Then $B$ is not closable (easy exercise from the definition), but $B$ is relatively $A$bounded with $A$bound zero. Hence, $A+B$ is closed (see Kato, Thorem IV.1.1). This example also answers your question. 


See Caradus's paper entitled: Semiclosed operators just the abstract or Messirdi's paper: Almost closed operator. 

