Stability of strongly continuous representations of abelian semigroups

Representations with regular norm-behaviour of locally compact abelian semigroups

“…The relation σ per (̺) ⊇ c ̺ σ per (ψ) implies that σ per (ψ) is also countable. Thus σ per (ψ) contains an eigenvalue χ of ψ * by [BV,Proposition 4.1]. Applying Theorem 23 again, we conclude that c ̺ χ ∈ σ p (̺ * ) ∩ {χ : |χ| = c ̺ }, which is a contradiction.…”

“…The conditions ensure that the associated isometric representation ψ of Theorem 23 acts on a non-zero Banach space Y. Thus σ per (ψ) is non-empty by [BV,Corollary 3.3]. Since c ̺ σ per (ψ) ⊂ σ per (̺), we infer that σ per (̺) is not empty.…”

“…If the conclusion fails to hold then by Theorem 23 the associated isometric representation ψ : S → L(Y) acts on a non-zero Banach space Y. Hence by [BV,Corollary 3.3], σ per (ψ) is not empty. The relation σ per (̺) ⊇ c ̺ σ per (ψ) implies that σ per (ψ) is also countable.…”

Ergodicity and stability of orbits of unbounded semigroup representations

“…of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700013598 [10] (…”

“…https://doi.org/10.1017/S1446788700013598 of Zarrabi [35]. The result for representations of groups is then used to prove global stability for semigroup representations T : J -> L(X) dominated by a weight w satisfying The key ingredient is [11, Proposition 3.1] which exploits a method developed by several authors (see [10,11,28 The authors thank the referee for his critical remarks.…”