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

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Conference Proceeding

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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).

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Proceedings of the American Mathematical Society

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