Abstract
We start with considering rank one self-adjoint perturbations A α = A +α( ⋅ , φ)φ with cyclic vector \(\varphi \in \mathcal{H}\) on a separable Hilbert space \(\mathcal{H}\). The spectral representation of the perturbed operator A α is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators A and A α .
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