OPERATORS OF GAMMA WHITE NOISE CALCULUS

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space ℱ_ext(ℋ) over ℋ= L²(ℝ^d, dσ) and show how to include the usual Fock space ℱ(ℋ) in it as a subspace. We introduce in ℱ_ext(ℋ) operators a(ξ)=∫_ℝ^ddxξ(x)a(x), ξ∈ S, with , where and ∂_x are the creation and annihilation operators at x. We show that (a(ξ))_ξ∈S is a family of commuting self-adjoint operators in ℱ_ext(ℋ) and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary I between ℱ_ext(ℋ) and the L²-space L²(S', dμ_G), where μ_G is the measure of Gamma white noise with intensity σ. The image of a(ξ) under I is the operator of multiplication by <·,ξ>, so that a(ξ)'s are Gamma field operators. The Fock structure of the Gamma space determined by I coincides with that discovered in Ref. 22. We note that I extends in a natural way the multiple stochastic integral (chaos) decomposition of the "chaotic" subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.