Some Remarks On Krein--von Neumann Extensions
Fritz Gesztesy, Selim Sukhtaiev
TL;DR
The paper surveys Krein--von Neumann extensions for strictly positive, densely defined operators with nonzero deficiency indices, focusing on resolvent properties of $S_K$ and the reduced operator $\widehat{S}_K$ and their relation to the Friedrichs extension $S_F$. It develops Krein-type resolvent formulas via Donoghue-type $M$-operators, including a linear fractional transformation linking $M$-functions of different extensions and a Krein-type resolvent formula between $S_K$ and $S_F$. It also presents a comprehensive parametrization of all nonnegative self-adjoint extensions via boundary data $(B,\mathcal{W})$ and analyzes domains, form methods, and spectral transfer, along with trace-ideal (Schatten) properties of resolvents. A key finding is that reduced resolvent information for $S_K$ implies certain trace-ideals for the Friedrichs block, but converses need not hold, as shown by Malamud’s results, highlighting limitations in deducing full spectral information from reduced data. The results illuminate the interplay between extremal extensions, operator-valued Nevanlinna–Herglotz functions, and trace-ideals in abstract spectral theory.
Abstract
We survey various properties of Krein--von Neumann extensions $S_K$ and the reduced Krein--von Neumann operator $\hatt S_K$ in connection with a strictly positive (symmetric) operator $S$ with nonzero deficiency indices. In particular, we focus on the resolvents of $S_K$ and $\hatt S_K$ and of the trace ideal properties of the resolvent of $\hatt S_K$, and make some comparisons with the corresponding properties of the resolvent of the Friedrichs extension $S_F$. We also recall a parametrization of all nonnegative self-adjoint extensions of $S$ and various Krein-type resolvent formulas for any two relatively prime self-adjoint extensions of $S$, utilizing a Donoghue-type $M$-operator (i.e., an energy parameter dependent Dirichlet-to-Neumann-type map).