In this paper we first construct a two-parameter transformation group G on the space of test white noise functionals in which the adjoints of Kuo＇s Fourier and Kuo＇s Fourier-Mehler transforms are included. Next we show that the group G is a two-dimensional complex Lie group whose infinitesimal generators are the Gross Laplacian Delta(G) and the number operator N, and then find an explicit description of a differentiable one-parameter subgroup of G whose infinitesimal generator is a Delta(G) + bN. As an application, we study the solution and fundamental solution for the Cauchy problem associated with a Delta(G) + bN. Finally we show that each element of the adjoint group G* of G can be characterized in terms of differentiation and multiplication operators.