  The functional linear model extends the notion of linear regression to the
case where the response and covariates are iid elements of an infinite
dimensional Hilbert space. The unknown to be estimated is a Hilbert-Schmidt
operator, whose inverse is by definition unbounded, rendering the problem of
inference ill-posed. In this paper, we consider the more general context where
the sample of response/covariate pairs forms a weakly dependent stationary
process in the respective product Hilbert space: simply stated, the case where
we have a regression between functional time series. We consider a general
framework of potentially nonlinear processes, exploiting recent advances in the
spectral analysis of time series. Our main result is the establishment of the
rate of convergence for the corresponding estimators of the regression
coefficients, the latter forming a summable sequence in the space of
Hilbert-Schmidt operators. In a sense, our main result can be seen as a
generalisation of the classical functional linear model rates, to the case of
time series, and rests only upon cumulant mixing conditions. While the analysis
becomes considerably more involved in the dependent case, the rates are
strikingly comparable to those of the i.i.d. case, but at the expense of an
additional factor caused by the necessity to estimate the spectral density
operator at a nonparametric rate, as opposed to the parametric rate for
covariance operator estimation.
