  This paper presents a new result on strong-consistency, in the trace norm, of
a diagonal componentwise parameter estimator of the autocorrelation operator of
an autoregressive process of order one (ARH(1) process), allowing
strong-consistency of the associated plug-in predictor. These results are
derived, when the eigenvectors of the autocovariance operator are unknown, and
the autocorrelation operator does not admit a diagonal spectral representation
with respect to the eigenvectors of the autocovariance operator.
