Characterisation of adjoint operators

Question

Let $X$ be a Banach space and $X^*$ denote his dual space. Then, it is well-known that if $T$ is a bounded linear operator on $X$, then $T^*$ is a bounded linear operator on $X^*$. My question is the following: Can we characterize the space $\{T^*: T\in\mathcal{L}(X)\}$ among $\mathcal{B}(X^*)$? Can it be the same space (apart from the finite dimensional case)?

Answer 1

A bounded operator on $X^*$ is the adjoint of a bounded operator on $X$ if and only if it is continuous for the weak* topology on $X^*$. This is still true if $X$ is a normed linear space, not necessarily complete. Since I'm answering the question I get to cite my book: see Corollary 4.9 of Measure Theory and Functional Analysis.

If $X$ is reflexive then every bounded operator on $X^*$ will be weak* continuous, and the converse is true too --- that's a nice exercise!