Boundedness and compactness of composition operators on Segal-Bargmann spaces

Abstract

For $E$ a Hilbert space, let $\mathcal{H}(E)$ denote the Segal-Bargmann space (also known as the Fock space) over $E$, which is a reproducing kernel Hilbert space with kernel $K(x,y)=\exp (<x,y>)$ for $x,y$ in $E$. If $\varphi$ is a mapping on $E$, the composition operator $C_{\varphi }$ is defined by $C_{\varphi }h=h\circ \varphi$ for $h\in \mathcal{H}(E)$ for which $h\circ \varphi$ also belongs to $\mathcal{H}(E)$. We determine necessary and sufficient conditions for the boundedness and compactness of $C_{\varphi }$. Our results generalize results obtained earlier by Carswell, MacCluer and Schuster for finite dimensional spaces $E$.

15 pages

References in corpus (1)

• On the isometric composition operators on the Bloch space in ${\mathbb{C}}^n$