Friedrichs and Kre\uın 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 $S_{\mathrm{f}}=S\widehat{+}(\{0\}\times \mathrm{mul}{S}^{\ast })$, the weak Friedrichs extension of $S$, and a selfadjoint extension $S_{\mathrm{F}}$, the Friedrichs extension of $S$, that satisfy $S\subset S_{\mathrm{f}}\subset S_{\mathrm{F}}$. The Friedrichs extension $S_{\mathrm{F}}$ 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 Kre\uın type extension $S_{\mathrm{k},c}=S\widehat{+}(\ker (S^{\ast }-c)\times \{0\})$ and Kre\uın type extension $S_{\mathrm{K},c}$ of $S$, that satisfy $S\subset S_{\mathrm{k},c}\subset S_{\mathrm{K},c}$. The Kre\uın type extension $S_{\mathrm{K},c}$ 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.

28 pages