Singular Perturbations as Range Perturbations in a Pontryagin Space

Abstract

When the singular finite rank perturbations of an unbounded selfadjoint operator Ao in a Hilbert space c, formally defined by A(α) = A 0 + GαG*, are lifted to an exit Pontryagin space ℌ by means of an operator model, they become ordinary range perturbations of a self-adjoint operator H ∞ in ℌ ⊃ ℌ0:H T = H ∞-ΩT -1Ω*. Here G is a mapping from ℂd into some scale space ℌ (A 0), ℕ, of generalized elements associated with A 0, while Ω is a mapping from into ℂd the extended space ℌ, where H T , is defined. The connection between these two perturbation formulas is studied.KeywordsSingular finite rank perturbationextension theoryKrein’s formulaboundary tripletWeyl functiongeneralized Nevanlinna functionoperator model