  The goal of this paper is to develop the theory of Schur complementation in
the context of operators acting on anti-dual pairs. As a byproduct, we obtain a
natural generalization of the parallel sum and parallel difference, as well as
the Lebesgue-type decomposition. To demonstrate how this operator approach
works in application, we derive the corresponding results for operators acting
on rigged Hilbert spaces, and for representable functionals of
${}^{*}$-algebras.
