Narrow Results

Let μ be a probability measure on I ⊂ ℝ with finite moment of all orders and γ_{α, c1, 2} be a probability measure on ℂ which is expressed by the modified Bessel function. In this paper, we shall first consider operators B^-, B⁺, B° and express them in terms of bosonic creation and annihilation operators. Secondly, we shall construct the Hilbert space of analytic L² functions with respect to γ_{α, c1, 2}, . It will be seen that the S_μ-transform under certain conditions is a unitary operator from L² (I, μ) onto . Moreover, we shall give explicit examples including Laguerre, Meixner and Meixner–Pollaczek polynomials and explain how the Hilbert space and B^-, B⁺, B° are related from the probabilistic point of view. It will also be given the relationship between the heat kernel on ℂ and γ_{α, c1, 2} (subordination property).