Gaussian kernel operators on white noise functional spaces

The Gaussian kernel operators on white noise functional spaces, including second quantization, Fourier-Mehler transform, scaling, renormalization, etc. are studied by means of symbol calculus, and characterized by the intertwining relations with annihilation and creation operators. The infinitesimal generators of the Gaussian kernel operators are second order white noise operators of which the number operator and the Gross Laplacian are particular examples.