Friedrichs and Kreĭn type extensions in terms of representing maps

Abstract

A semibounded operator or relation S in a Hilbert space with lower bound \(\gamma \in {{\mathbb {R}}}\) has a symmetric extension \(S_\textrm{f}=S \, \widehat{+} \,(\{0\} \times \mathrm{mul\,}S^*)\), the weak Friedrichs extension of S, and a selfadjoint extension \(S_{\textrm{F}}\), the Friedrichs extension of S, that satisfy \(S \subset S_{\textrm{f}} \subset S_\textrm{F}\). The Friedrichs extension \(S_{\textrm{F}}\) has lower bound \(\gamma \) and it is the largest semibounded selfadjoint extension of S. Likewise, for each \(c \le \gamma \), the relation S has a weak Kreĭn type extension \(S_{\textrm{k},c}=S \, \widehat{+} \,(\mathrm{ker\,}(S^*-c) \times \{0\})\) and Kreĭn type extension \(S_{\textrm{K},c}\) of S, that satisfy \(S \subset S_{\textrm{k},c} \subset S_{\textrm{K},c}\). The Kreĭn type extension \(S_{\textrm{K},c}\) has lower bound c and it is the smallest semibounded selfadjoint extension of S which is bounded below by c. In this paper these special extensions and, more generally, all extremal extensions of S are constructed via the semibounded sesquilinear form \({{\mathfrak {t}}}(S)\) that is associated with S; the representing map for the form \({{\mathfrak {t}}}(S)-c\) plays an essential role here.