  We consider strictly stationary stochastic processes of Hilbert space-valued
random variables and focus on fully functional tests for the equality of the
lag-zero autocovariance operators of several independent functional time
series. A moving block bootstrap-based testing procedure is proposed which
generates pseudo random elements that satisfy the null hypothesis of interest.
It is based on directly bootstrapping the time series of tensor products which
overcomes some common difficulties associated with applications of the
bootstrap to related testing problems. The suggested methodology can be
potentially applied to a broad range of test statistics of the hypotheses of
interest. As an example, we establish validity for approximating the
distribution under the null of a test statistic based on the Hilbert-Schmidt
distance of the corresponding sample lag-zero autocovariance operators, and
show consistency under the alternative. As a prerequisite, we prove a central
limit theorem for the moving block bootstrap procedure applied to the sample
autocovariance operator which is of interest on its own. The finite sample size
and power performance of the suggested moving block bootstrap-based testing
procedure is illustrated through simulations and an application to a real-life
dataset is discussed.
