Let (E)βandbe the space of test and generalized white noise functionals, respectively. It is proved that for each positive integerkwe construct explicitly one-parameter semigroup and cosine family of operators on an appropriate (E)βspace of which infinitesimal generator is. As an application, we will prove the existence and uniqueness of solutions of the Cauchy problems for the first and second order differential equations associated with the operator. Here △GandNare the Gross Laplacian and the number operator on (E)β, respectively. Also, we investigate the existence and uniqueness of solutions of the Cauchy problems associated with the operatoris the adjoint operator of △G