Boundedness and compactness of composition operators on Segal-Bargmann spaces

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 $ϕ$ is a mapping on $E$, the composition operator $C_ϕ$ is defined by $C_ϕh = h\circϕ$ for $h\in \mathcal{H}(E)$ for which $h\circϕ$ also belongs to $\mathcal{H}(E)$. We determine necessary and sufficient conditions for the boundedness and compactness of $C_ϕ$. Our results generalize results obtained earlier by Carswell, MacCluer and Schuster for finite dimensional spaces $E$.

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