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

Operators on anti-dual pairs: self-adjoint extensions and the Strong Parrott Theorem