Sum of Closed Operators Closable?
On $\ell^2$, define $A$ and $B$ by $(Ax)_n = -(Bx)_n = n^2 x_n$ for $n > 1$, $(A x)_1 = \sum_{n=1}^\infty n x_n$ and $(B x)_1 = 0$, with $D(A) = D(B) = \{x: \sum_{n =1}^\infty n^4 |x_n|^2 < \infty \}$. Then if I'm not mistaken $A$ and $B$ are closed but $A + B$ is not closable, e.g. (with $e_n$ the standard unit vectors) $\lim_{n \to \infty} e_n/n =0$ while $(A + B) e_n/n = e_1$.