General linear processes in Hilbert spaces and prediction

We discuss a general definition of linear processes in Hilbert spaces that takes into account the outstanding role played by this model in prediction theory.Actually this definition is based on the Wold decomposition of a weakly stationary process (Xn)(Xn) with values in a Hilbert space H. It leads to processes of the formXn=εn+∑j=1∞λj(εn-j),n∈Z,where (εn)(εn) is a H  -white noise and (λj)(λj) a sequence of (possibly) unbounded linear operators. A necessary and sufficient condition for boundedness of λjλj is given.As applications we introduce and study general H-autoregressive processes, H-moving average processes and H-Markov processes in the wide sense.Specific examples are given.