Let $(X,\|\cdot\|)$ be a Banach space and $(X^*,\|\cdot\|_{op})$ its continuous dual Banach space.
Let $\text{End}_{op}(X^*)$ be the continuous linear endomorphisms of $X^*$ with respect to $\|\cdot\|_{op}$ and $\text{End}_*(X^*)$ the continuous linear endomorphisms with respect to the weak-* topology of $X^*$.
Does $\text{End}_{op}(X^*)=\text{End}_*(X^*)$? If not when are they equal and how do they compare in general? Also references and proofs are welcome.
