We consider a positive self-adjoint operator A and formal rank one perturbations where φ ∈ −2(A) but φ ∉ −1 (A), with s(A) the usual scale of spaces. We show that B can be defined for such φ and what are essentially negative infinitesimal values of α. In a sense we will make precise, every rank one perturbation is one of three forms: (i) φ ∈ −1(A), α ∈ ; (ii) φ ∈ −1, α = ∞; or (iii) the new type we consider here.
