  In this paper, we propose a new test for the equality of several covariance
functions for functional data. Its test statistic is taken as the supremum
value of the sum of the squared differences between the estimated individual
covariance functions and the pooled sample covariance function, hoping to
obtain a more powerful test than some existing tests for the same testing
problem. The asymptotic random expression of this test statistic under the null
hypothesis is obtained. To approximate the null distribution of the proposed
test statistic, we describe a parametric bootstrap method and a non-parametric
bootstrap method. The asymptotic random expression of the proposed test is also
studied under a local alternative and it is shown that the proposed test is
root-$n$ consistent. Intensive simulation studies are conducted to demonstrate
the finite sample performance of the proposed test and it turns out that the
proposed test is indeed more powerful than some existing tests when functional
data are highly correlated. The proposed test is illustrated with three real
data examples.
