  We provide square-root n consistency results regarding estimation of the
spectral representation of covariance operators of Hilbertian time series, in a
setting with imperfect measurements. This is a generalization of the method
developed in Bathia et al. (2010). The generalization relies on an important
property of centered random elements in a separable Hilbert space, namely, that
they lie almost surely in the closed linear span of the associated covariance
operator. We provide a straightforward proof to this fact. This result is, to
our knowledge, overlooked in the literature. It incidentally gives a rigorous
formulation of PCA in Hilbert spaces.
