Given any closed, densely defined operator $T$ in $B$, is it then true that there exists a closed, densely defined operator $\tilde T$ in $H_2$ which extends $T$ (viewed as an operator in $H_2$)? In other words, is $T$ closable as an operator in $H_2$?
In a paper by Gill, Basu, Zachary and Steadman, it is pointed out that it is well known that the answer to my question is 'yes' (See Theorem 4). They argue that the adjoint of $T$, say $T':B'\rightarrow B'$, is a closed operator, which is densely defined if we restrict it to $H_2'\subset B'$. However, it is not explained why the restriction of $T'$ is densely defined, and it does not seem obvious to me that it is.
This question is related to another question of mine, Pointwise approximation of a closed operator.