## On the class of extremal extensions of a nonnegative operator.(English)Zbl 0996.47029

Kérchy, László (ed.) et al., Recent advances in operator theory and related topics. The Béla Szőkefalvi-Nagy memorial volume. Proceedings of the memorial conference, Szeged, Hungary, August 2-6, 1999. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 127, 41-81 (2001).
Let $$A$$ be a densely defined nonnegative operator in a separable Hilbert space $$\mathfrak H$$ with inner product $$(\cdot, \cdot)$$. There exist two nonnegative selfadjoint extensions, the Friedrichs extension $$A_{F}$$ and the Krejn-von Neumann extension $$A_{N}$$, with the property that every nonnegative selfadjoint extension $$\widetilde{A}$$ of $$A$$ satisfies the inequalities $((A_{F}-x)^{-1}h,h)\leq ((\widetilde{A}-x)^{-1}h,h) \leq ((A_{N}-x)^{-1}h,h) \text{ for $$h\in \mathfrak H$$, $$x\leq 0$$}.$ A nonnegative selfadjoint extension $$\widetilde{A}$$ of $$A$$ is called extremal if $$\inf \{(\widetilde{A} (\varphi-f),\varphi-f): f\in \text{dom } A\}=0$$ for all $$\varphi\in \text{dom} \widetilde{A}$$.
The paper obtains a construction of extremal extensions, descriptions of extremal extensions via the Friedrichs and the Krejn-von Neumann extensions, and also obtains a purely analytic description of extended extensions. Moreover the paper obtains some applications of these results to the second differential operator $$-D^{2}$$ on $$[0,1]$$, a one-dimensional Schrödinger operator on the halfline and a one-dimensional Schrödinger operator with point-interactions.
For the entire collection see [Zbl 0971.00017].

### MSC:

 47B25 Linear symmetric and selfadjoint operators (unbounded) 47A20 Dilations, extensions, compressions of linear operators 47A63 Linear operator inequalities 47B65 Positive linear operators and order-bounded operators