Arkiv för Matematik

, Volume 43, Issue 2, pp 365–382 | Cite as

Distance near the origin between elements of a strongly continuous semigroup

  • Jean Esterle
Article

Abstract

Set
$$\theta (s/t): = (s/t - 1)(t/s)^{\frac{{s/t}}{{s/t - 1}}} = (s - t)\frac{{t^{t/(s - t)} }}{{s^{s/(s - t)} }}$$
if 0<t<s. The key result of the paper shows that if (T (t))t>0 is a nontrivial strongly continuous quasinilpotent semigroup of bounded operators on a Banach space then there exists δ>0 such that ║T(t)-T(s)║>θ(s/t) for 0<t<s≤δ. Also if (T(t))t>0 is a strongly continuous semigroup of bounded operators on a Banach space, and if there exists η>0 and a continuous functionts(t) on [0, ν], satisfyings(0)=0, and such that 0<t<s(t) and ║T(t)-T(s(t))║<θ(s/t) fort∈(o, η], then the infinitesimal generator of the semigroup is bounded. Various examples show that these results are sharp.

Keywords

Banach Space Bounded Operator Continuous Semigroup Infinitesimal Generator 

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Copyright information

© Institut Mittag-Leffler 2005

Authors and Affiliations

  • Jean Esterle
    • 1
  1. 1.Laboratoire d'Analyse et Géométrie UMR 5467Université Bordeaux 1TalenceFrance

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