High order singular rank one perturbations of a positive operator

In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression Lα = L + α〈·, φ〉φ are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space H with inner product 〈· ,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to H−n \ H−n+1 with n ≥ 3, where {Hs}∞s=−∞ is the scale of Hilbert spaces associated with L in H