Completeness of rank one perturbations of normal operators with lacunary spectrum

Abstract

Suppose $\mathcal{A}$ is a compact normal operator on a Hilbert space $H$ with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let $\mathcal{L}$ be its rank one perturbation. We show that either infinitely many moment equalities hold or the linear span of root vectors of $\mathcal{L}$, corresponding to non-zero eigenvalues, is of finite codimension in $H$. In contrast to classical results, we do not assume that the perturbation is weak.

22 pages

References in corpus (1)

• Recent developments in spectral synthesis for exponential systems and for non-self-adjoint operators

Cited by in corpus (1)

• Localization of zeros in Cauchy-de Branges spaces