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A class of bilateral weighted shift operators, and linear dynamics

Bibhash Kumar Das, Aneesh Mundayadan

Abstract

This article aims to initiate a study of bilateral weighted backward shift operators defined on the spaces $\ell^p_{a,b}(Ω_{r,R})$ and $c_{0,a,b}(Ω_{r,R})$ 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 $B_w$.

A class of bilateral weighted shift operators, and linear dynamics

Abstract

This article aims to initiate a study of bilateral weighted backward shift operators defined on the spaces and which are Banach spaces of analytic functions on a suitable annulus in the complex plane, having a normalized Schauder basis of the form, We obtain necessary and sufficient conditions for a weighted shift to be bounded, and find conditions so that is similar to a compact perturbation of a weighted shift on . In addition, we study when is hypercyclic, supercyclic, and chaotic. It shown that the zero-one law of orbital limit points does not hold for , which is in contrast to the case of weighted shifts on . Most of our results are obtained using the matrix form of .
Paper Structure (7 sections, 24 theorems, 112 equations)

This paper contains 7 sections, 24 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. The spaces $\ell^p_{a,b}(\Omega_{r,R})$ and $c_{0,a,b}(\Omega_{r,R})$
  3. Continuity of coefficient functionals
  4. Weighted backward shift $B_w$: matrix forms, and consequences
  5. $B_w$ as compact + usual shifts, and the essential spectrum
  6. Hypercyclicity, supercyclicity, and chaos
  7. Orbital limit points versus hypercyclicity

Key Result

Proposition 2.1

For $1\leq p\leq \infty,$ and $\{\lambda_n\}\in \ell^p(\mathbb{Z})$, the series converges uniformly and absolutely on compact subsets of $\Omega_{r,R}$. Moreover, the representation of $f(z)$ in terms of $\{\lambda_n\}\in \ell^p(\mathbb{Z})$ and $\{f_n\}$ is unique. The same conclusions hold true if we take $\{\lambda_n\}$ from $c_0(\mathbb{Z})$.

Theorems & Definitions (44)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 34 more