A central limit theorem for operators

Abstract

We prove an analogue of the Central Limit Theorem for operators. For every operator K defined on C[x] we construct a sequence of operators KN defined on C[x1,...,xN] given by the N-fold tensor product of K with itself (modulo a proper scaling) and we demonstrate that, under certain orthogonality conditions, this sequence converges in a weak sense to an unique operator C. We show that this operator C is a member of a family of operators C that we call 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 [1] 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.