Title: Semigroup dynamics for flight vectors

Authors: Robert E. O'Brien

Addresses: Department of Mathematics(Retired), The Catholic University of America, Washington, DC 20064, USA

Abstract: A commutative semigroup of contractions on a Hilbert space, , has a natural order and resultant net structure which defines stability, system dynamics, and α and ω limits for the flight vectors ₀. 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. ₀ splits: ₀ = _m ⊕ _w, _w the weakly stable subspace and _m the weakly Poisson recurrent space. _m = ⊕M(x_τ, ) where M(x_τ, ) is the closed subspace spanned by the weak limit points of x_τ, {x_τ} an orthonormal set of weakly-wandering vectors in _m. Examples illustrate the results.

Keywords: Hilbert space; recurrence; flight vectors; dynamical system; weakly wandering; almost weakly stable; unitary; weak stability; contraction; semigroup.

DOI: 10.1504/IJDSDE.2020.109108

International Journal of Dynamical Systems and Differential Equations, 2020 Vol.10 No.4, pp.358 - 372

Received: 19 Jul 2018
Accepted: 21 Dec 2018
Published online: 20 Aug 2020 *

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