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 $A_λ= A + Σ_{k=1}^n λ_k(\cdot,ϕ_k)ϕ_k$ is pure point for a. e. $λ=(λ_1,λ_2,...,λ_n) \in\Bbb R^n$ iff operator $A_η=A+η(\cdot,ϕ_k)ϕ_k$ is pure point for a.e.\ $η\in\Bbb R$ for $k=1,2,...,n$. We also show that if $A_λ$ is pure point for a.e.\ $λ\in \Bbb R^n$ then $A_λ$ is pure point for a.e.\ $λ\in γ$ for any analytic curve $γ\in\Bbb R^n$.