Semigroup dynamics for flight vectors (Q2019899)

Summary: A commutative semigroup of contractions \(\mathcal{S}\) on a Hilbert space, \(\mathcal{H}\), has a natural order and resultant net structure which defines stability, system dynamics, and \(\alpha\) and \(\omega\) limits for the flight vectors \(\mathcal{H}_0\). The space of `pure' flight vectors -- no nontrivial weakly stable components -- are spanned by the \(\omega\) limits of \textit{weakly-wandering} vectors which are \textit{weakly Poisson recurrent}. \(H_0\) splits: \(\mathcal{H}_0 = \mathcal{H}_m \oplus \mathcal{H}_w\), \(\mathcal{H}_w\) the \textit{weakly stable} subspace and \(H_m\) the weakly \textit{Poisson recurrent} space. \( \mathcal{H}_m = \oplus M(x_\tau, \mathcal{S})\) where \(M(x_\tau, \mathcal{S})\) is the closed subspace spanned by the weak limit points of \(x_\tau \), \(\{x_\tau\} \) an orthonormal set of \textit{weakly-wandering} vectors in \(\mathcal{H}_m\). Examples illustrate the results.