  The aim of this paper is to develop an approach to obtain self-adjoint
extensions of symmetric operators acting on anti-dual pairs. The main advantage
of such a result is that it can be applied for structures not carrying a
Hilbert space structure or a normable topology. In fact, we will show how
hermitian extensions of linear functionals of involutive algebras can be
governed by means of their induced operators. As an operator theoretic
application, we provide a direct generalization of Parrott's theorem on
contractive completion of $2$ by $2$ block operator-valued matrices. To exhibit
the applicability in noncommutative integration, we characterize hermitian
extendibility of symmetric functionals defined on a left ideal of a
$C^{*}$-algebra.
