Semigroup dynamics for flight vectors

Abstract

A commutative semigroup of contractions S on a Hilbert space, ℌ, has a natural order and resultant net structure which defines stability, system dynamics, and α and ω limits for the flight vectors ℌ0. The space of 'pure' flight vectors - no nontrivial weakly stable components - are spanned by the ω limits of weakly-wandering vectors which are weakly Poisson recurrent. ℌ0 splits: ℌ0 = ℌm ⊕ ℌw, ℌw the weakly stable subspace and Hm the weakly Poisson recurrent space. ℌm = ⊕M(xτ, S) where M(xτ, S) is the closed subspace spanned by the weak limit points of xτ, {xτ} an orthonormal set of weakly-wandering vectors in ℌm. Examples illustrate the results.