Ultra Feller operators from a functional-analytic perspective

It is widely known 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 to be a variant of the theorem that on a Banach space with the Dunford–Pettis property, the product of two weakly compact operators is compact.