On unconditional basisness of B-quasi-exponentials

TL;DR

This paper investigates conditions under which the eigenfunctions of a perturbed quasinilpotent operator form an unconditional basis, establishing the necessity of the Muckenhoupt (A2) condition without spectrum restrictions.

Contribution

It proves the necessity of the Muckenhoupt (A2) condition for unconditional basisness of eigenfunctions of a perturbed quasinilpotent operator, extending previous results without spectrum restrictions.

Findings

Necessity of Muckenhoupt (A2) condition established

Unconditional basisness linked to operator perturbation

No restrictions on the spectrum of the operator

Abstract

We consider a quasinilpotent operator whose resolvent is entire operator function of exponential type. Let A be its one-dimensional perturbation. We establish necessity of Muckenhoupt condition (A2) for two weights related to operator A for unconditional basisness of its eigenfunctions. Note that no apriori restrictions are imposed on the spectrum of A.