Distances entre exponentielles et puissances d’éléments de certaines algèbres de Banach

Abstract

Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and letWe also show, assuming a suitable spectral condition on x, that if x ≥ 1 − 1 (γ + 1)

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 ,…”

“…and it follows from Lemma 2.4 that r(t) = tr (1). We therefore have JT (t) = e tv for t > 0 with v = u + r(1).…”

“…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.…”

Dichotomy Results for Norm Estimates in Operator Semigroups

“…One particular case of the above is used in the estimates considered by Bendaoud, Esterle and Mokhtari [2,10]. Corollary 2.10 Let γ > 0 and 0 < α < π/2.…”

“…these include µ = δ 1 −δ 2 , the difference of two Dirac measures, where F (s) := Lµ(s) = e −s − e −2s and F (−sA) = T (t) − T (2t). More importantly, the theorem applies to many other examples, such as dµ(t) = (χ [1,2] − χ [2,3] )(t)dt and µ = δ 1 − 3δ 2 + δ 3 + δ 4 , which are not accessible with the methods of [1,9,10,14].…”