Let $X$ be a Banach space and let $A\in B(X^*)$. I want to know if there exists some operator $B\in B(X)$ such that $B^*=A$, that is if the "adjoint" operation in surjective. Thank you !
$A = B^*$ for some $B$ if and only if $A$ is continuous in the weak-* topology. See e.g. the answer to this question. One of the comments there gives an example of an $A$ that is not continuous in the weak-* topology.