  We perform an in-depth study of some domination and smoothing properties of
linear operators and of their role within the theory of eventually positive
operator semigroups. On the one hand we prove that, on many important function
spaces, they imply compactness properties. On the other hand, we show that
these conditions can be omitted in a number of Perron--Frobenius type spectral
theorems. We furthermore prove a Kre\u{\i}n--Rutman type theorem on the
existence of positive eigenvectors and eigenfunctionals under certain eventual
positivity conditions.
