Abstract

Several results concerning the spaces ( E) and ( E)* of test and generalized white noise functionals, respectively, are obtained. The irreducibility of the canonical commutation relation for operators on ( E) and on ( E)* is proved. It is shown that the Fourier-Mehler transform F 0 on ( E)* is the adjoint of a continuous linear operator G 0 on ( E). Moreover, a characterization theorem for the Fourier-Mehler transform is proved. In particular, the Fourier transform is the unique (up to a constant) continuous linear operator F on ( E)* such that F D̃ ξ = q̃ ξ F and F q̃ ξ = − D̃ ξ F . Here D̃ ξ and q̃ ξ are differentiation and multiplication operators, respectively. Several one-parameter transformation groups acting on ( E) and the Lie algebra generated by their infinitesimal generators are also discussed.

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