Title
Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures
Document Type
Conference Proceeding
Publication Date
3-1-2003
Abstract
Let μg and μp denote the Gaussian and Poisson measures on ℝ, respectively. We show that there exists a unique measure μ̃g on C such that under the Segal-Bargmann transform Sμg the space L2(ℝ, μg) is isomorphic to the space HL2 (ℂ, μ̃g) of analytic L2-functions on ℂ with respect to μ̃g. We also introduce the Segal-Bargmann transform Sμp for the Poisson measure μp and prove the corresponding result. As a consequence, when μg and μp have the same variance, L2(ℝ, μg) and L2(ℝ, μp) are isomorphic to the same space HL2(ℂ,μ̃g) under the Sμg-and Sμp-transforms, respectively. However, we show that the multiplication operators by x on L2(ℝ,μg) and on L2(ℝ,μp) act quite differently on HL2(ℂ, μ̃g).
Publication Source (Journal or Book title)
Proceedings of the American Mathematical Society
First Page
815
Last Page
823
Recommended Citation
Asai, N., Kubo, I., & Kuo, H. (2003). Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures. Proceedings of the American Mathematical Society, 131 (3), 815-823. https://doi.org/10.1090/S0002-9939-02-06564-4