Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung

doi:10.1090/S0002-9939-02-06564-4

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures
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by Nobuhiro Asai, Izumi Kubo and Hui-Hsiung Kuo PDF

Proc. Amer. Math. Soc. 131 (2003), 815-823 Request permission

Abstract:

Let $\mu _{g}$ and $\mu _{p}$ denote the Gaussian and Poisson measures on ${\mathbb R}$, respectively. We show that there exists a unique measure $\widetilde {\mu }_{g}$ on ${\mathbb C}$ such that under the Segal-Bargmann transform $S_{\mu _g}$ the space $L^2({\mathbb R},\mu _g)$ is isomorphic to the space ${\mathcal H}L^2({\mathbb C}, \widetilde {\mu }_{g})$ of analytic $L^2$-functions on ${\mathbb C}$ with respect to $\widetilde {\mu }_{g}$. We also introduce the Segal-Bargmann transform $S_{\mu _p}$ for the Poisson measure $\mu _{p}$ and prove the corresponding result. As a consequence, when $\mu _{g}$ and $\mu _{p}$ have the same variance, $L^2({\mathbb R},\mu _g)$ and $L^2({\mathbb R},\mu _p)$ are isomorphic to the same space ${\mathcal H}L^2({\mathbb C}, \widetilde {\mu }_{g})$ under the $S_{\mu _g}$- and $S_{\mu _p}$-transforms, respectively. However, we show that the multiplication operators by $x$ on $L^2({\mathbb R}, \mu _g)$ and on $L^2({\mathbb R}, \mu _p)$ act quite differently on ${\mathcal H}L^2({\mathbb C}, \widetilde {\mu }_{g})$.

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