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

Authors: Nobuhiro Asai, Izumi Kubo and Hui-Hsiung Kuo
Journal: Proc. Amer. Math. Soc. 131 (2003), 815-823
MSC (2000): Primary 46L53; Secondary 33D45, 44A15
Published electronically: July 2, 2002
MathSciNet review: 1937419
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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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Additional Information

Nobuhiro Asai
Affiliation: International Institute for Advanced Studies, Kizu, Kyoto, 619-0225, Japan
Address at time of publication: Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

Izumi Kubo
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

Hui-Hsiung Kuo
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Keywords: Interacting Fock space, Segal-Bargmann transform, coherent vector, Gaussian measure, Poisson measure, space of square integrable analytic functions, decomposition of multiplication operator
Received by editor(s): August 18, 2001
Received by editor(s) in revised form: October 12, 2001
Published electronically: July 2, 2002
Additional Notes: Research of the first author supported by a Postdoctoral Fellowship of the International Institute for Advanced Studies, Kyoto, Japan
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2002 American Mathematical Society