A convolution semigroup plays an important role in the theory of probability measure on Lie groups. The basic problem is that one wants to express a semigroup as a Lévy-Khinchine formula. If $(\mu_t)_{t\in\mathbf{R}_+^*}$ is a continuous semigroup of probability measures on a Hilbert-Lie group $G,$ then we define
$$T_{\mu_t}f:=\int f_a\mu_t(da)\quad (f\in C_*(G); \ t > 0).$$
It is apparent that $(T_{\mu_t})_{t\in\mathbf{R}_+^*}$ is a continuous operator semigroup on the space $C_*(G)$ with the infinitesimal generator $N$. The generating functional $A$ of this semigroup is defined by $Af := \lim_{t\downarrow 0}\frac 1t (T_{\mu_t}f(e)-f(e))$. We consider the problem of construction of a subspace $C_{(2)}(G)$ of $C_*(G)$ such that the generating functional $A$ on $C_{(2)}(G)$ exists. This result will be used later to show that Lvy-Khinchine formula holds for Hilbert-Lie groups.