Rank one perturbations of diagonal operators without eigenvalues

TL;DR

This paper demonstrates that certain diagonal operators with a perfect spectrum can be perturbed with a rank one operator to eliminate eigenvalues, addressing a specific open question in operator theory.

Contribution

It proves that diagonal operators of multiplicity one with perfect spectrum can be perturbed by rank one operators to produce eigenvalue-free operators, answering Ionascu's question.

Findings

Existence of eigenvalue-free rank one perturbations for specified diagonal operators

Resolution of Ionascu's open question in operator theory

Advancement in understanding spectral properties under perturbations

Abstract

In this paper, we prove that every diagonal operator on a Hilbert space of which is of multiplicity one and has perfect spectrum admits a rank one perturbation without eigenvalues. This answers a question of Ionascu.