## 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.
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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 ${\mathcal H}L^2({\mathbb C}, \widetilde {\mu }_{g})$.## References

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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