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 $A_\lambda= A + \Sigma_{k=1}^n \lambda_k(\cdot,\phi_k)\phi_k$ is pure point for a. e. $\lambda=(\lambda_1,\lambda_2,...,\lambda_n) \in\Bbb R^n$ iff operator $A_\eta=A+\eta(\cdot,\phi_k)\phi_k$ is pure point for a.e.\ $\eta\in\Bbb R$ for $k=1,2,...,n$. We also show that if $A_\lambda$ is pure point for a.e.\ $\lambda\in \Bbb R^n$ then $A_\lambda$ is pure point for a.e.\ $\lambda\in \gamma$ for any analytic curve $\gamma\in\Bbb R^n$.