  In this paper we investigate the question of estimating the Gram operator by
a robust estimator from an i.i.d. sample in a separable Hilbert space and we
present uniform bounds that hold under weak moment assumptions. The approach
consists in first obtaining non-asymptotic dimension-free bounds in
finite-dimensional spaces using some PAC-Bayesian inequalities related to
Gaussian perturbations of the parameter and then in generalizing the results in
a separable Hilbert space. We show both from a theoretical point of view and
with the help of some simulations that such a robust estimator improves the
behavior of the classical empirical one in the case of heavy tail data
distributions.
