  We derive and study a significance test for determining if a panel of
functional time series is separable. In the context of this paper, separability
means that the covariance structure factors into the product of two functions,
one depending only on time and the other depending only on the coordinates of
the panel. Separability is a property which can dramatically improve
computational efficiency by substantially reducing model complexity. It is
especially useful for functional data as it implies that the functional
principal components are the same for each member of the panel. However such an
assumption must be verified before proceeding with further inference. Our
approach is based on functional norm differences and provides a test with well
controlled size and high power. We establish our procedure quite generally,
allowing one to test separability of autocovariances as well. In addition to an
asymptotic justification, our methodology is validated by a simulation study.
It is applied to functional panels of particulate pollution and stock market
data.
