Segal-Bargmann Transforms of One-mode Interacting Fock Spaces Associated with Gaussian and Poisson Measures

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dc.creatorAsai, Nobuhiro
dc.creatorKubo, Izumi
dc.creatorKuo, Hui-Hsiung
dc.date2001-10-01
dc.date.accessioned2026-07-07T04:43:35Z
dc.date.available2026-07-07T04:43:35Z
dc.descriptionLet $μ_{g}$ and $μ_{p}$ denote the Gaussian and Poisson measures on ${\Bbb R}$, respectively. We show that there exists a unique measure $\widetildeμ_{g}$ on ${\Bbb C}$ such that under the Segal-Bargmann transform $S_{μ_g}$ the space $L^2({\Bbb R},μ_g)$ is isomorphic to the space ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$ of analytic $L^2$-functions on ${\Bbb C}$ with respect to $\widetildeμ_{g}$. We also introduce the Segal-Bargmann transform $S_{μ_p}$ for the Poisson measure $μ_{p}$ and prove the corresponding result. As a consequence, when $μ_{g}$ and $μ_{p}$ have the same variance, $L^2({\Bbb R},μ_g)$ and $L^2({\Bbb R},μ_p)$ are isomorphic to the same space ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$ under the $S_{μ_g}$ and $S_{μ_p}$-transforms, respectively. However, we show that the multiplication operators by $x$ on $L^2({\Bbb R}, μ_g)$ and on $L^2({\Bbb R}, μ_p)$ act quite differently on ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$.
dc.descriptionLouisiana State University Preprint No. 2001-14
dc.identifierhttps://arxiv.org/abs/math/0110011
dc.identifierhttp://arxiv.org/abs/math/0110011
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/62290
dc.subjectProbability
dc.subject46L53, 33D45, 44A15
dc.titleSegal-Bargmann Transforms of One-mode Interacting Fock Spaces Associated with Gaussian and Poisson Measures
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