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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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:
  • Izumi Kubo
  • Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan
  • Email:
  • Hui-Hsiung Kuo
  • Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
  • Email:
  • 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:
  • MathSciNet review: 1937419