  Let $W$ and $Z$ be Banach spaces such that $Z$ is separable and let
$R:W\longrightarrow Z$ be a (continuous, linear) operator. We study
consequences of the adjoint operator $R^\ast$ having non-separable range. From
our main technical result we obtain applications to the theory of basic
sequences and the existence of universal operators for various classes of
operators between Banach spaces. We also obtain an operator-theoretic
characterisation of separable Banach spaces with non-separable dual.
