On Rank One Perturbations of Selfadjoint Operators

Let A be a selfadjoint operator in a Hilbert space h. Its rank one perturbations A+τ(·,ω)ω, τ ∈ R, are studied when ω belongs to the scale space h-2 associated with h+2 = dom A and (·,·) is the corresponding duality. If A is nonnegative and ω belongs to the scale space h-1, it is proven that the spectral measures of A(τ), τ ∈ R, converge weakly to the spectral measure of the limiting perturbation A(∞). In fact A(∞) can be identified as a Friedrichs extension. Further results for nonnegative operators A were obtained by allowing ω ∈ h-2. Our purpose is to show that most of these results 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-called Q-functions facilitates the descriptions. In the special case that ω belongs to the scale space h-1 associated with h+1 = dom |A|^½, the limiting perturbation A(∞) is shown to be the generalized Friedrichs extension.