Recurrence Operators on Function Spaces

TL;DR

This paper investigates recurrence properties of linear operators in various function spaces, including Banach algebras and weighted Dirichlet spaces, revealing that certain composition operators are never recurrent.

Contribution

It extends the study of recurrence to Banach algebras and composition operators with linear fractional symbols, providing new insights into their dynamical behavior.

Findings

Composition operators with parabolic non automorphism symbols are not recurrent.

Recurrence properties are characterized for operators on Banach algebras and weighted Dirichlet spaces.

The work connects recurrence with hypercyclicity in the context of composition operators.

Abstract

Although recurrence for dynamical systems has been studied since the end of the nineteenth century, the study of recurrence for linear operators started with papers by Costakis, Manoussos and Parissis in 2012 and 2014. We explore recurrence in Banach algebras, in the space of continuous linear operators on $\omega=\mathbb{C}^{\mathbb{N}},$ and for composition operators whose symbols are linear fractional transformations, acting on weighted Dirichlet spaces. In particular, we show that composition operators with a parabolic non automorphism symbol are never recurrent. Our results relate to work by Gallardo-Guti\'errez and Montes-Rodr\'iguez in their 2004 AMS Memoir in which they show, among other things, that composition operators induced by parabolic non automorphisms are never hypercyclic in weighted Dirichlet spaces. Our work also relates to a more recent paper by Karim, Benchiheb, and Amouch.