I asked this question on Math StackExchange first, but it was not answered. 


If $X$ is a Banach space and $Z$ is a subset of $X^\*$, consider the annihilator of $Z$ in $X^{\*\*}$:


$$
 Z^{\perp}=\{x^{\*\*}\in X^{\*\*} : x^{\*\*}(Z)=0\}
$$


and the pre-anihilator of $Z$ in $X$:


$$
 Z^{\top}=\{x\in X : y^\*(x)=0, \forall y^\*\in Z\}
$$


It is easy to see that $Z^\top\subseteq Z^{\perp}$ when the elements of $X$ are viewed as functionals on $X^\*$ via the canonical embedding. It is also clear that when $X$ is reflexive we have equality. My question: is there a characterization of those subsets $Z$ of $X^\*$ that give equality in the non-reflexive case as well? 



Annihilators and pre-annihilators
As I understand the conditions are: (1) The closed linear span of $Z$ in $X^\*$ is weak$^\*$ closed; (2) The subspace $Z^\top$ in $X$ is reflexive.


