We consider a probability measure m m on a Hilbert space X X and a bounded linear transformation on X X that preserves the measure. We characterize the linear dynamical systems ( X , m , T ) (X,m,T) for the cases where either X X is finite dimensional or T T is unitary and we give an example where T T is a weighted shift operator. We apply the results to the limit identification problem for a vector-valued ergodic theorem of A. Beck and J. T. Schwartz, n − 1 ( Σ i n T i F i ) → F ¯ {n^{ - 1}}(\Sigma _i^n{T^i}{F_i}) \to \overline F a.s., where F i {F_i} is a stationary sequence of integrable X X -valued random variables and T T a unitary operator on X X .