CANONICAL SYSTEMS AND FINITE RANK PERTURBATIONS OF SPECTRA

ALEXEI G. POLTORATSKI

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 ndimensional case, we prove that for any cyclic self-adjoint operator A, operator A_{λ} = A + Σ_{k=1}^{n}λ_{k}(·, ϕ_{k})ϕ_{k} is pure point for a. e. λ = (λ_{1}, λ_{2}, ..., λ_{n}) ∈ ℝ^{n} iff operator A_{η} = A + η(·, ϕ_{k})ϕ_{k} is pure point for a.e. η ∈ ℝ for k = 1, 2, ..., n. We also show that if A_{λ} is pure point for a.e. λ ∈ ℝ^{n} then A_{λ} is pure point for a.e. λ ∈ γ for any analytic curve γ ∈ ℝ^{n}.