Constructing one-parameter transformations on white noise functions in terms of equicontinuous generators

Let X be a barreled locally convex space. A continuous operator Ξ on X is called an equicontinuous generator if {Ξⁿ / n = 0,1,2,...} is an equicontinuous family of operators. For each equicontinuous generator a one-parameter group of operators is constructed by means of power series There is a one-to-one correspondence between the equicontinuous generators and the locally equicontinuous holomorphic one-parameter groups of operators. If two equicontinuous generators Ξ 1 Ξ 2 satisfy [Ξ, Ξ 2] = αΞ 2 for some α ∈ ℂ then aΞ1 + bΞ 2 is also an equicontinuous generator for any a, b ∈ ℂ. These general results are applied to a study of operators on white noise functions. In particular, a linear combination of the number operator and the Gross Laplacian, which are natural infinite dimensional analogues of a finite dimensional Laplacian, is always an equicontinuous generator. This result contributes to the Cauchy problems in white noise (Gaussian) space.