Cyclic Composition Operators on Segal-Bargmann space

TL;DR

This paper investigates the cyclic, supercyclic, and hypercyclic properties of affine composition operators on the Segal-Bargmann space, providing characterizations of boundedness and how the symbol's properties affect cyclic behavior.

Contribution

It offers a new characterization of symbols inducing bounded composition operators and links the properties of these symbols to the cyclic behavior of the operators.

Findings

Characterization of symbols for bounded composition operators

Conditions under which $C_{}$ is cyclic, supercyclic, or hypercyclic

Influence of the symbol's properties on cyclic behavior

Abstract

We study the hypercyclic, supercyclic and cyclic properties of composition operator $C_{\phi}$ on the Segal-Bargmann space $\mathscr{H}(\mathscr{E})$, where $\phi (z)=Az+b$, $A\in \mathcal{B}(\mathscr{E})$, $b\in \mathscr{E}$ with $\left\|A\right\|\leq 1$ and $A^*b\in (I-A^*A)^{\frac{1}{2}}$. In this connection we also give a characterization of the symbols $\phi$ which induce the bounded composition operator $C_{\phi}$ on $\mathscr{H}(\mathscr{E})$ and show that the properties of $\phi$ influence the cyclic behaviour of $C_{\phi}$.