Some Remarks On Krein--von Neumann Extensions

We survey various properties of Krein–von Neumann extensions $S_K$ and the reduced Krein–von Neumann operator $\text{hatt}{S}_K$ in connection with a strictly positive (symmetric) operator $S$ with nonzero deficiency indices. In particular, we focus on the resolvents of $S_K$ and $\text{hatt}{S}_K$ and of the trace ideal properties of the resolvent of $\text{hatt}{S}_K$, 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).