Let $P_sφ(x)=\mathbb{E}\, φ(X^x(s))$, be the transition semigroup on the space $B_b(E)$ of bounded measurable functions on a Banach space $E$, of the Markov family defined by the linear equation with additive noise $$ d X(s)= \left(AX(s) + a\right)ds + BdW(s), \qquad X(0)=x\in E. $$ We give a simple probabilistic proof of the fact that null-controlla\-bility of the corresponding deterministic system $$ d Y(s)= \left(AY(s)+ B\mathcal{U}(t)x)(s)\right)ds, \qquad Y(0)=x, $$ implies that for any $φ\in B_b(E)$, $P_tφ$ is infinitely many times Fréchet differentiable and that $$ D^nP_tφ(x)[y_1,\ldots ,y_n]= \mathbb{E}\, φ(X^x(t))(-1)^nI^n_t(y_1,\ldots, y_n), $$ where $I^n_t(y_1,\ldots,y_n)$ is the symmetric n-fold Itô integral of the controls $\mathcal{U}(t)y_1,\ldots \mathcal{U}(t)y_n$.