On the spectrum of weighted shifts

TL;DR

This paper investigates the spectrum of weighted shift operators in linear dynamics, focusing on the point spectrum and spectrum under certain regularity conditions on the weight sequence.

Contribution

It provides new insights into the spectral properties of weighted shifts, especially relating to their point spectrum and spectrum under regularity assumptions.

Findings

Characterization of the point spectrum of weighted shifts

Conditions under which the spectrum can be deduced

Connections between spectral properties and weight sequence regularity

Abstract

It is well-known that, in Linear Dynamics, the most studied class of linear operators is certainly that of weighted shifts, on the separable Banach spaces $c_0$ and $\ell^p$, $1 \leq p< \infty$. Over the last decades, the intensive study of such operators has produced an incredible number of versatile, deep and beautiful results, applicable in various areas of Mathematics and the relationships between various important notions, especially concerning chaos and hyperbolic properties, and the spectrum of weighted shifts are investigated. In this paper, we investigate the point spectrum of weighted shifts and, under some regularity hypotheses on the weight sequence, we deduce the spectrum.