  The problem of comparing the entire second order structure of two functional
processes is considered and a $L^2$-type statistic for testing equality of the
corresponding spectral density operators is investigated. The test statistic
evaluates, over all frequencies, the Hilbert-Schmidt distance between the two
estimated spectral density operators. Under certain assumptions, the limiting
distribution under the null hypothesis is derived. A novel frequency domain
bootstrap method is introduced, which leads to a more accurate approximation of
the distribution of the test statistic under the null than the large sample
Gaussian approximation derived. Under quite general conditions, asymptotic
validity of the bootstrap procedure is established for estimating the
distribution of the test statistic under the null. Furthermore, consistency of
the bootstrap-based test under the alternative is proved. Numerical simulations
show that, even for small samples, the bootstrap-based test has a very good
size and power behavior. An application to a bivariate real-life functional
time series illustrates the methodology proposed.
