Linear dynamical systems

Abstract:

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 theorem of A. Beck and J. T. Schwartz, ${n^{ - 1}}(\Sigma _i^n{T^i}{F_i}) \to \overline F$ a.s., where ${F_i}$ is a stationary sequence of integrable $X$-valued random variables and $T$ a unitary operator on $X$.