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


$$\theta (s/t): = (s/t - 1)(t/s)^{\frac{{s/t}}{{s/t - 1}}} = (s - t)\frac{{t^{t/(s - t)} }}{{s^{s/(s - t)} }}$$
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.


Banach Space Bounded Operator Continuous Semigroup Infinitesimal Generator 
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.


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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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