  We present a general theory to quantify the uncertainty from imposing
structural assumptions on the second-order structure of nonstationary Hilbert
space-valued processes, which can be measured via functionals of time-dependent
spectral density operators. The second-order dynamics are well-known to be
elements of the space of trace-class operators, the latter is a Banach space of
type 1 and of cotype 2, which makes the development of statistical inference
tools more challenging. A part of our contribution is to obtain a weak
invariance principle as well as concentration inequalities for (functionals of)
the sequential time-varying spectral density operator. In addition, we
introduce deviation measures in the nonstationary context, and derive
estimators that are asymptotically pivotal. We then apply this framework and
propose statistical methodology to investigate the validity of structural
assumptions for nonstationary response surface data, such as low-rank
assumptions in the context of time-varying dynamic fPCA and principle separable
component analysis, deviations from stationarity with respect to the square
root distance, and deviations from zero functional canonical coherency.
