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.

中文翻译：

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反对偶对的算子：自伴扩展和强帕罗特定理

本文的目的是开发一种方法来获得作用于反对偶对的对称算子的自伴随扩展。这种结果的主要优点是它可以应用于不携带希尔伯特空间结构或规范拓扑的结构。事实上，我们将展示对合代数的线性泛函的厄密扩展如何通过它们的诱导算子来控制。作为算子理论应用，我们提供了 Parrott 定理的直接推广，即 $2$ 块运算符值矩阵的 $2$ 的收缩完成。为了展示在非交换积分中的适用性，我们描述了定义在 $C^{*}$-代数的左理想上的对称泛函的厄密扩展性。