Uniqueness of form extensions and domination of semigroups

Abstract

In this article, we study questions of uniqueness of form extension for certain magnetic Schr\"odinger forms. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. We review this concept in an abstract setting and give a characterization in terms of the associated forms. Then we use it to prove a theorem that transfers uniqueness of form extension of a dominating form to that of a dominated form. This result is applied in two concrete situations: magnetic Schr\"odinger forms on graphs and on domains in Euclidean space.