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 \({\mathcal{H}}.\)
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Dijksma, A., Kurasov, P. & Shondin, Y. High Order Singular Rank One Perturbations of a Positive Operator. Integr. equ. oper. theory 53, 209–245 (2005). https://doi.org/10.1007/s00020-005-1357-5
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DOI: https://doi.org/10.1007/s00020-005-1357-5