Completeness of rank one perturbations of normal operators with lacunary spectrum

TL;DR

This paper investigates the completeness of root vectors for rank one perturbations of compact normal operators with lacunary spectra, revealing conditions under which the span of these vectors is either finite-codimensional or satisfies infinitely many moment equalities.

Contribution

It establishes new criteria for the completeness of root vectors in rank one perturbations of normal operators with lacunary spectra without assuming weak perturbations.

Findings

Either infinitely many moment equalities hold or the root vectors span a finite-codimensional subspace.

The spectral eigenvalues decay exponentially fast due to lacunarity.

Completeness properties depend on the spectral distribution and perturbation characteristics.

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.