Hypercyclicity of operators that $\lambda$-commute with the Hardy backward shift

TL;DR

This paper investigates the dynamics of operators that $ ext{lambda}$-commute with the Hardy backward shift, extending the understanding of hypercyclicity in the context of operator commutation relations.

Contribution

It introduces the study of hypercyclicity for operators that $ ext{lambda}$-commute with the Hardy backward shift, expanding prior characterizations to this broader class.

Findings

Operators $X$ that $ ext{lambda}$-commute with $B$ can be hypercyclic under certain conditions.

The paper generalizes known results about the commutant of the Hardy backward shift.

New criteria for hypercyclicity of $ ext{lambda}$-commuting operators are established.

Abstract

An operator $T$ acting on a separable complex Hilbert space $H$ is said to be hypercyclic if there exists $f\in H$ such that the orbit $\{T^n f:\ n\in \mathbb{N}\}$ is dense in $H$. Godefroy and Shapiro \cite{GoSha} characterized those elements in the commutant of the Hardy backward shift which are hypercyclic. In this paper we study some dynamics properties of operators $X$ that $\lambda$-commute with the Hardy backward shift $B$, that is, $BX=\lambda XB$.