  The problem of testing equality of the entire second order structure of two
independent functional linear processes is considered. A fully functional
$L^2$-type test is developed which evaluates, over all frequencies, the
Hilbert-Schmidt distance between the estimated spectral density operators of
the two processes. The asymptotic behavior of the test statistic is
investigated and its limiting distribution under the null hypothesis is
derived. Furthermore, a novel frequency domain bootstrap method is developed
which approximates more accurately the distribution of the test statistic under
the null than the large sample Gaussian approximation obtained. Asymptotic
validity of the bootstrap procedure is established and consistency of the
bootstrap-based test under the alternative is proved. Numerical simulations
show that, even for small samples, the bootstrap-based test has very good size
and power behavior. An application to meteorological functional time series is
also presented.
