  The assumption of separability of the covariance operator for a random image
or hypersurface can be of substantial use in applications, especially in
situations where the accurate estimation of the full covariance structure is
unfeasible, either for computational reasons, or due to a small sample size.
However, inferential tools to verify this assumption are somewhat lacking in
high-dimensional or functional {data analysis} settings, where this assumption
is most relevant. We propose here to test separability by focusing on
$K$-dimensional projections of the difference between the covariance operator
and a nonparametric separable approximation. The subspace we project onto is
one generated by the eigenfunctions of the covariance operator estimated under
the separability hypothesis, negating the need to ever estimate the full
non-separable covariance. We show that the rescaled difference of the sample
covariance operator with its separable approximation is asymptotically
Gaussian. As a by-product of this result, we derive asymptotically pivotal
tests under Gaussian assumptions, and propose bootstrap methods for
approximating the distribution of the test statistics. We probe the finite
sample performance through simulations studies, and present an application to
log-spectrogram images from a phonetic linguistics dataset.
