Transformations for White Noise Functionals

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 F0 on (E)* is the adjoint of a continuous linear operator G0 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 FD̃ξ = q̃ξF and Fq̃ξ = − D̃ξF. Here D̃ξ and q̃ξ 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.