Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Nobuhiro Asai Izumi Kubo Hui-Hsiung Kuo

Institution:	International Institute for Advanced Studies, Kizu, Kyoto, 619-0225, Japan ; Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan ; Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

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 -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 on $L^2({\mathbb R}, \mu_g)$ and on $L^2({\mathbb R}, \mu_p)$ act quite differently on ${\mathcal H}L^2({\mathbb C}, \widetilde{\mu}_{g})$ .

Keywords:	Interacting Fock space Segal-Bargmann transform coherent vector Gaussian measure Poisson measure space of square integrable analytic functions decomposition of multiplication operator

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