High Order Singular Rank One Perturbations of a Positive Operator

Abstract.

In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression $L_{\alpha} = L + \alpha\langle \cdot , \varphi \rangle \varphi$ are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space ${\mathcal{H}}$ with inner product 〈·,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to $\mathcal{H}_{{ - n}} \backslash \mathcal{H}_{{ - n + 1}}$ with n ≥ 3, where ${\left\{ {\mathcal{H}_{s} } \right\}}^{\infty }_{{s = - \infty }}$ is the scale of Hilbert spaces associated with L in ${\mathcal{H}}.$