On rank one perturbations of selfadjoint operators

Abstract

LetA be a selfadjoint operator in a Hilbert space ℌ. Its rank one perturbations A+τ(·,ω)ω, ℝ, are studied when ω belongs to the scale space ℌ−2 associated with ℌ+2=domA and (·,·) is the corresponding duality. IfA is nonnegative and ω belongs to the scale space ℌ−1, Gesztesy and Simon [4] prove that the spectral measures ofA(τ), ℝ, converge weakly to the spectral measure of the limiting perturbationA(∞). In factA(∞) can be identified as a Friedrichs extension. Further results for nonnegative operatorsA were obtained by Kiselev and Simon [14] by allowing ω∈ℌ−2, Our purpose is to show that most results of Gesztesy, Kiselev, and Simon are valid for rank one perturbations of selfadjoint operators, which are not necessarily semibounded. We use the fact that rank one perturbations constitute selfadjoint extensions of an associated symmetric operator. The use of so-calledQ-functions [6, 8] facilitates the descriptions. In the special case that ω belongs to the scale space ℌ−1 associated with ℌ+2=dom |A|1/2 the limiting perturbationA(∞) is shown to be the generalized Friedrichs extension [5].