Singular Rank One Perturbations of Self-Adjoint Operators and Krein Theory of Self-Adjoint Extensions

Abstract

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A_∞,ω for A_α,ω = A + α (·ω)ω,α→∞ where A is a self-adjoint positive operator, ω∈$\mathcal{H}_{ - 1} (\mathcal{H}_s ,\;s \in R^1$ being the ‘A-scale’). In the present note it is remarked that the operator A_∞,ω also appears directly as the Friedrichs extension of the symmetric operator $\dot A$:=A⌈ \{f∈$\mathcal{D}$(A)| 〈f,ω〉=0\}. It is also shown that Krein's resolvents formula: (A_{_b,ω}-z)^-1 =(A-z)^-1+$b_z^{ - 1}$ (·,$\eta _{\bar z}$) η_z, with b=b-(1+z) (η_z,η_-1),η_z= (A-z)^-1ω defines a self-adjoint operator A_b,ω for each ω∈$\mathcal{H}_{ - 2}$ and b∈ R¹. Moreover it is proven that for any sequence ω_n∈ $\mathcal{H}_{ - 1}$ which goes to ω in $\mathcal{H}_{ - 2}$ there exists a sequence α_n→0 such that $A_{\alpha _n ,\omega _n }$ → A_b,ω in the strong resolvent sense.