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

For selfadjoint extensions of a symmetric densely defined positive operator Amin, the lower boundedness problem is the question of whether is lower bounded if and only if an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension Aγ has compact inverse (Grubb, 1974, also Gorbachuk and Mikhailets, 1976); this applies to elliptic operators A on bounded domains.For exterior domains, is not compact, and whereas the lower bounds satisfy , the implication of lower boundedness from T to has only been known when m(T)>−m(Aγ). 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+ε)/2m) for t→∞.