Canonical systems and finite rank perturbations of spectra

Abstract

We use Rokhlin's Theorem on the uniqueness of canonical systems to find a new way to establish connections between Function Theory in the unit disk and rank one perturbations of self-adjoint or unitary operators. In the n-dimensional case, we prove that for any cyclic self-adjoint operator $A$, operator $A_{\lambda }=A+{\Sigma }_{k=1}^n{\lambda }_k(\cdot ,{\varphi }_k){\varphi }_k$ is pure point for a. e. $\lambda =({\lambda }_1,{\lambda }_2,...,{\lambda }_n)\in {\mathbb{R}}^n$ iff operator $A_{\eta }=A+\eta (\cdot ,{\varphi }_k){\varphi }_k$ is pure point for a.e.\ $\eta \in \mathbb{R}$ for $k=1,2,...,n$. We also show that if $A_{\lambda }$ is pure point for a.e.\ $\lambda \in {\mathbb{R}}^n$ then $A_{\lambda }$ is pure point for a.e.\ $\lambda \in \gamma$ for any analytic curve $\gamma \in {\mathbb{R}}^n$.

References in corpus (1)

• The Krein spectral shift and rank one perturbations of spectra