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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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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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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
  • Email: asai@kurims.kyoto-u.ac.jp
  • Izumi Kubo
  • Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan
  • Email: kubo@math.sci.hiroshima-u.ac.jp
  • Hui-Hsiung Kuo
  • Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
  • Email: kuo@math.lsu.edu
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 815-823
  • MSC (2000): Primary 46L53; Secondary 33D45, 44A15
  • DOI: https://doi.org/10.1090/S0002-9939-02-06564-4
  • MathSciNet review: 1937419