  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.
