Frequently hypercyclic weighted backward shifts on spaces of real analyticfunctions

Berkay Anahtarci,Can Deha Kariksiz

doi:10.3906/mat-1806-81

Abstract

Frequently hypercyclic weighted backward shifts on spaces of real analyticfunctions

Highlights

Let A(Ω) denote the space of real analytic functions on a nonempty open subset Ω of R, equipped with its natural topology of inductive limit of the spaces H(U ) of holomorphic functions on U, where U runs through all complex open neighborhoods of Ω
We define weighted backward shifts on A(Ω) as in [8] by considering their action on monomials using the density of polynomials in A(Ω)
In this paper we focus on the linear dynamical property called frequent hypercyclicity, introduced by Bayart and Grivaux [1], for which an orbit meets every nonempty open set frequently in terms of positive lower density

Summary

Introduction

Let A(Ω) denote the space of real analytic functions on a nonempty open subset Ω of R , equipped with its natural topology of inductive limit of the spaces H(U ) of holomorphic functions on U , where U runs through all complex open neighborhoods of Ω . the spaces H(U ) are Fréchet with the topology of uniform convergence on compact subsets, and monomials form a (Schauder) basis for H(U ), the space A(Ω) is not metrizable and does not admit a Schauder basis [9]. We recall that a continuous linear operator T on a topological vector space X is hypercyclic if there is a vector x in X such that the set {T nx : n ∈ N} , called the orbit of x under T , is dense in X . Certain conditions on hypercyclicity and chaoticity of weighted backward shifts on spaces of real analytic functions were given in [8].

Results

Conclusion