Friedrichs and Kre\u{\i}n type extensions in terms of representing maps

Abstract

A semibounded operator or relation $S$ in a Hilbert space with lower bound $m \in {\mathbb R}$ has a symmetric extension $S_{\rm f}=S {\, \widehat + \,} (\{0\} \times {\rm mul\,} S^*)$, the weak Friedrichs extension of $S$, and a selfadjoint extension $S_{\rm F}$, the Friedrichs extension of $S$, that satisfy $S \subset S_{\rm f} \subset S_{\rm F}$. The Friedrichs extension $S_{\rm F}$ has lower bound $\gamma$ and it is the largest semibounded selfadjoint extension of $S$. Likewise, for each $c \leq \gamma$, the relation $S$ has a weak Kre\u{\i}n type extension $S_{{\rm k},c}=S {\, \widehat + \,} (\ker (S^*-c) \times \{0\})$ and Kre\u{\i}n type extension $S_{{\rm K},c}$ of $S$, that satisfy $S \subset S_{{\rm k},c} \subset S_{{\rm K},c}$. The Kre\u{\i}n type extension $S_{{\rm K},c}$ 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 in terms of a representing map for ${\mathfrak t}(S)-c$ and their properties are being considered.