We characterize the adjoint operator ranges of the dual of a Banach space E, that is, the subspaces of \(E^*\) that are the range of the adjoint of a bounded linear operator between E and another Banach space. We also show that the space \(E^*\) contains an adjoint proper dense operator range if, and only if, there exists a weak*-closed subspace Z of \(E^*\) such that \(E^*/Z\) is infinite-dimensional and separable (which is equivalent to the existence of an infinite-dimensional Asplund subspace of E). As an application, we obtain a characterization of Asplund-saturated Banach spaces. Finally, we provide some results concerning the existence of quasicomplementary subspaces with a prescribed behavior with respect to adjoint proper dense operator ranges in dual spaces.
