DISTANCE ENTRE ÉlÉMENTS D'UN SEMI-GROUPE DANS UNE ALGÉBRE DE BANACH

Abstract

Soit ðat Þ t>0 un semi-groupe dans une alge`bre de Banach, et soit n un entier positif.

Distances between elements of a semigroup and estimates for derivatives

“…The results we shall derive are stronger forms of those of Bendaoud, Esterle and Mokhtari [1,5], in the case of semigroups defined on the positive real line. For example, they show that if lim sup t→0 T (t) − T ((n + 1)t) < n (n + 1) 1+1/n ,…”

“…In fact, there are two cases to consider, the easier non-quasinilpotent case, and then the general case. Further, we show that, unlike the results of [5], these results cannot be extended to a result using only T (t) for t in any proper measurable subgroup of R (for example, Q).…”

“…The following result is a strengthening of Theorem 2.3 of [5]: Theorem 2.1 Let (T (t)) t>0 be a non-quasinilpotent semigroup in a Banach algebra, let A be the closed subalgebra generated by (T (t)) t>0 and let γ > 0 be a real number. If…”

“…Bendaoud, Esterle and Mokhtari [1,5]. The third author showed in [4] that if (T (t)) t>0 is a nontrivial strongly continuous quasinilpotent semigroup, then there exists δ > 0 such that we have, for 0 < t < s ≤ δ, It is easy to deduce from this result that tT (t) ≥ 1 e for t ∈ (0, δ] if (T (t)) t>0 is a quasinilpotent differentiable semigroup.…”

“…Proof : Let φ ∈ A; then, by Lemma 2.1 of [5] there exists c ∈ C such that φ(T (t)) = e ct for all t > 0. We have…”