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 L (R, μg) is isomorphic to the space HL(C, μ̃g) of analytic L-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, L(R, μg) and L(R, μp) are isomorphic to the same space HL(C, μ̃g) under the Sμg and Sμp -transforms, respectively. However, we show that the multiplication operators by x on L(R, μg) and on L(R, μp) act quite differently on HL(C, μ̃g).