Let $X$ and $Y$ be two Banach spaces. Let $T : Y^* \rightarrow X^*$ be a unbounded linear map where $X^*$ represents the Banach dual of $X$.
My question is, there exists a dense domain $D \subseteq X$ and a linear map $S : D \rightarrow Y$ such that $$ \langle Sx , y \rangle = \langle x, Ty \rangle $$ for any $x \in D$, $y \in Y^*$? (Please, read the bra-ket notation as a duality product). Are $S$ or $D$ unique?
Motivation: I want to prove that a map $T : Y^* \rightarrow X^*$ is $w^*$-continuous, and I know a `candidate' $S:X \rightarrow Y$ for being the preadjoint is bounded on norm. The only doubt that remains for me is if there exists an unbounded preadjoint for $T$, because if not, I am afraid I am doing void reasonings...
Thank you