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.