A class of bilateral weighted shift operators, and linear dynamics

TL;DR

This paper studies bilateral weighted shift operators on specific Banach spaces of analytic functions, establishing conditions for boundedness, similarity to compact perturbations, and exploring their dynamic properties like hypercyclicity and chaos.

Contribution

It introduces a new class of bilateral weighted shift operators on analytic function spaces and characterizes their boundedness, similarity, and dynamical behaviors.

Findings

Characterized boundedness conditions for the operators.

Identified when operators are similar to compact perturbations.

Analyzed hypercyclicity, supercyclicity, and chaos of the operators.

Abstract

This article aims to initiate a study of bilateral weighted backward shift operators defined on the spaces $\ell^p_{a,b}(\Omega_{r,R})$ and $c_{0,a,b}(\Omega_{r,R})$ which are Banach spaces of analytic functions on a suitable annulus in the complex plane, having a normalized Schauder basis of the form, $$ f_n(z):= (a_n+b_{n}z)z^{n},\hskip 0.5cm n\in \mathbb{Z}. $$ We obtain necessary and sufficient conditions for a weighted shift $B_w$ to be bounded, and find conditions so that $B_w$ is similar to a compact perturbation of a weighted shift on $\ell^p(\mathbb{Z})$. In addition, we study when $B_w$ is hypercyclic, supercyclic, and chaotic. It shown that the zero-one law of orbital limit points does not hold for $B_w$, which is in contrast to the case of weighted shifts on $\ell^p(\mathbb{Z})$. Most of our results are obtained using the matrix form of $B_w$.