Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures
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
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