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].


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