Ultra Feller operators from a functional analytic perspective

It is a widely acknowledged fact that the product of two positive strong Feller operators on a Polish space $E$ enjoys the ultra Feller property. We present a functional analytic proof of this fact that allows us to drop the assumption that the operators are positive and also extends the applicability of this result to more general state spaces. As it turns out, this result can be considered a variant of the theorem that on a Banach space with the Dunford--Pettis property, the product of two weakly compact operators is compact.