  We propose a stable version of Principal Component Analysis (PCA) in the
general framework of a separable Hilbert space. It consists in interpreting the
projection on the first eigenvectors as a step function applied to the spectrum
of the covariance operator and in replacing it with a smooth cut-off of the
eigenvalues. We study the problem from a statistical point of view, so that we
assume that we do not have direct access to the covariance operator but we have
to estimate it from an i.i.d. sample. We provide some results on the quality of
the approximation of our spectral cut-off in terms of the quality of the
approximation of the eigenvalues of the covariance operator.
