(n) (x) · v). The map F is called a linear cocycle. Fix some norm ·· on R d , and use the same symbol to indicate the induced operator norm.

We consider a probability measure m on a Hilbert space X and a bounded linear transformation on X that preserves the measure. We characterize the linear dynamical systems (X, m, T) for the cases where either X is finite dimensional or T is unitary and we give an example where T is a weighted shift operator. We apply the results to the limit identification problem for a vector-valued ergodic the...