Uniqueness of form extensions and domination of semigroups

In this article, we present a new method to study uniqueness of form extensions in a rather general setting. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. Our main abstract result transfers uniqueness of form extension of a dominating form to that of a dominated form. This result can be applied to a multitude of examples including various magnetic Schr\"odinger forms on graphs and on manifolds.
Comment: The previous version of this article has been split. This version contains the material of the second part, the first part can be found at arXiv:1711.07225. This article is partly based on the third named author's master's thesis (arXiv:1604.05114)