  This paper addresses the fundamental task of estimating covariance matrix
functions for high-dimensional functional data/functional time series. We
consider two functional factor structures encompassing either functional
factors with scalar loadings or scalar factors with functional loadings, and
postulate functional sparsity on the covariance of idiosyncratic errors after
taking out the common unobserved factors. To facilitate estimation, we rely on
the spiked matrix model and its functional generalization, and derive some
novel asymptotic identifiability results, based on which we develop DIGIT and
FPOET estimators under two functional factor models, respectively. Both
estimators involve performing associated eigenanalysis to estimate the
covariance of common components, followed by adaptive functional thresholding
applied to the residual covariance. We also develop functional information
criteria for the purpose of model selection. The convergence rates of estimated
factors, loadings, and conditional sparse covariance matrix functions under
various functional matrix norms, are respectively established for DIGIT and
FPOET estimators. Numerical studies including extensive simulations and two
real data applications on mortality rates and functional portfolio allocation
are conducted to examine the finite-sample performance of the proposed
methodology.
