Linear Dynamical Systems

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 ~ l(2,"T'Fj) -> F a.s., where Ff is a stationary sequence of integrable Xvalued random variables and T a unitary operator on X. A dynamical system is defined as a triplet {X, m, T), whereA is a separable metric space, m is a probability measure on the Borel a-algebra of X and 7: X —»X is a continuous measure preserving transformation (m.p.t.). If, in addition, A' is a linear topological vector space and T a continuous linear transformation, then the dynamical system will be called linear. Linear dynamical systems can serve as models for representing stationary stochastic sequences as orbit sequences under T of linear functionals on X. In this study we take A to be a separable complex Hilbert space. Definitions, (i) A complex number A is called an eigenvalue of the m.p.t. 7 if there exists a complex valued measurable functions/(•) on X such that /(7(-)) = A/(-) a.e. It follows that the eigenvalues form a countable subgroup of the circle group, (ii) the m.p.t. 7 is called ergodic if