  This paper presents new results on prediction of linear processes in function
spaces. The autoregressive Hilbertian process framework of order one (ARH(1)
process framework) is adopted. A componentwise estimator of the autocorrelation
operator is formulated, from the moment-based estimation of its diagonal
coefficients, with respect to the orthogonal eigenvectors of the
auto-covariance operator, which are assumed to be known. Mean-square
convergence to the theoretical autocorrelation operator, in the space of
Hilbert-Schmidt operators, is proved. Consistency then follows in that space.
For the associated ARH(1) plug-in predictor, mean absolute convergence to the
corresponding conditional expectation, in the considered Hilbert space, is
obtained. Hence, consistency in that space also holds. A simulation study is
undertaken to illustrate the finite-large sample behavior of the formulated
componentwise estimator and predictor. The performance of the presented
approach is compared with alternative approaches in the previous and current
ARH(1) framework literature, including the case of unknown eigenvectors.
