  We investigate the concept of cylindrical Wiener process subordinated to a
strictly $\alpha$-stable L\'evy process, with $\alpha\in\left(0,1\right)$, in
an infinite dimensional, separable Hilbert space, and consider the related
stochastic convolution. We then introduce the corresponding Ornstein-Uhlenbeck
process, focusing on the regularizing properties of the Markov transition
semigroup defined by it. In particular, we provide an explicit, original
formula -- which is not of Bismut-Elworthy-Li's type -- for the Gateaux
derivatives of the functions generated by the operators of the semigroup, as
well as an upper bound for the norm of their gradients. In the case
$\alpha\in\left(\frac{1}{2},1\right)$, this estimate represents the starting
point for studying the Kolmogorov equation in its mild formulation.
