On the dynamics of a general unitary operator

A unitary operator U acting on the weakly compact unit ballB of a Hilbert space H gives a topological dynamical system (B, U). We prove in this note that for every ergodic U-invariant measure m, the dynamical system (B, U, m) is conjugated to a strictly ergodic group translation. Especially, the topological entropy of (B, U) is zero. As an application, it follows that any flow on the unit ball given by a Schrödinger equation ihψ = Lψ has zero topological entropy.