Krein-like extensions and the lower boundedness problem for elliptic operators

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For selfadjoint extensions tilde-A of a symmetric densely defined positive operator Aₘin, the lower boundedness problem is the question of whether tilde-A is lower bounded if and only if an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension Agamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets 1976); this applies to elliptic operators A on bounded domains. For exterior domains, Agamma ⁻¹ is not compact, and whereas the lower bounds satisfy m(T)≥ m(tilde-A), the implication of lower boundedness from T to tilde-A has only been known when m(T)>-m(Agamma). We now show it for general T. The operator Aₐ corresponding to T=aI, generalizing the Krein-von Neumann extension A₀, appears here; its possible lower boundedness for all real a is decisive. We study this Krein-like extension, showing for bounded domains that the discrete eigenvalues satisfy N+(t;Aₐ)=cₐtⁿ/²ᵐ+O(t(n-1+varepsilon)/2m) for t→∞.