If $X$ is a Banach space and $Z$ is a subspace 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 subspaces (subsets ?) $Z$ of $X^*$ that give equality in the non-reflexive case as well?