  We prove an analogue of the Central Limit Theorem for operators. For every
operator $K$ defined on $\mathbb{C}[x]$ we construct a sequence of operators
$K_N$ defined on $\mathbb{C}[x_1,...,x_N]$ and demonstrate that, under certain
orthogonality conditions, this sequence converges in a weak sense to an unique
operator $\mathcal{C}$. We show that this operator $\mathcal{C}$ is a member of
a family of operators $\mathfrak{C}$ that we call {\it Centered Gaussian
Operators} and which coincides with the family of operators given by a centered
Gaussian Kernel. Inspired in the approximation method used by Beckner in [W.
Beckner, Inequalities in Fourier Analysis, Annals of Mathematics, 102 (1975),
159-182] to prove the sharp form of the Hausdorff-Young inequality, the present
article shows that Beckner's method is a special case of a general
approximation method for operators. In particular, we characterize the Hermite
semi-group as the family of Centered Gaussian Operators associated with any
semi-group of operators.
