A General Theory of Sufficient Collections of Norms with a Prescribed Semigroup of Contractions
Nahum Zobin, Veronica Zobina · Birkhäuser Basel eBooks · 1994
The paper contains a complete exposition and proofs of our results on the theory of sufficient collections of norms. Consider a uniformly bounded irreducible semigroup G of operators in a finite dimensional real linear space V . We consider G -contractive norms on V , i.e., such norms that every operator from G is contractive with respect to each of them. A collection of G -contractive norms { ∥ · ∥ α } α∊ A is called sufficient if for any linear operator T : V → V the following implication holds: if T is contractive in every norm ‖ · ‖ α ( α ∊ A ) then T is contractive in every G -contractive norm. We describe all sufficient collections, construct two canonical sufficient collections and study their extremal properties. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.