Hypercyclic Functions for Backward and Bilateral Shift Operators

Z.A Hassanain,N Faried,A Morsy

doi:10.3844/jmssp.2009.178.182

Abstract

Problem statement: Giving conditions for bilateral forward and unilateral backward shift operators over the weighted space of p-summable formal series to be hypercyclic. This provides a generalization to the case of Hilbert space. Approach: We used hypercyclicity criterion and some preliminary concepts for formal Laurent series and formal power series. Moreover we got benefits of some duality properties of above mentioned spaces. Results: We obtained necessary and sufficient conditions for bilateral forward and unilateral backward shift operators to be hypercyclic. Conclusion: The bilateral forward shift operator was hypercyclic on the space of all formal Laurent series and the unilateral backward shift operator was hypercyclic on the space of all formal power series under certain conditions.

Highlights

A vector x in a Banach space X is called hypercyclic for a bounded operator T if the orbit
The first examples of hypercyclic operators appeared in the space of entire functions defined over the complex plane, endowed with the compact-open topology
The notion of hypercyclicity on Banach spaces started in 1969 with Rolewicz[3], who showed that any scalar multiple λ B of the unilateral backward shift B is hypercyclic on lp (1 ≤ p 1

Summary

Introduction

A vector x in a Banach space X is called hypercyclic for a bounded operator T if the orbit{ } Tnx : n ≥ 0 is dense in X. Kitai in his thesis with title invariant closed sets for linear operators, university of Toronto, determined conditions under which a linear operator is hypercyclic. This result, commonly referred to as the hypercyclicity criterion, was never published and a few years later it was rediscovered in a broader form by Gethner and Shapiro[5]. During the last year's hypercyclicity criterion on Banach or Frechet spaces has attracted many mathematicians working in linear functional analysis and very important contributions to the topic have been made[5,6,7,8,9,10,11,12,13]

Methods

Discussion

Conclusion