Cyclic Composition Operators on Segal-Bargmann space

Abstract

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

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