Arkiv för Matematik

  • Ark. Mat.
  • Volume 43, Number 2 (2005), 365-382.

Distance near the origin between elements of a strongly continuous semigroup

Jean Esterle

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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 function ts(t) on [0, ν], satisfying s(0)=0, and such that 0< t< s(t) and ║T(t)-T(s(t))║<θ(s/t) for t∈(o, η], then the infinitesimal generator of the semigroup is bounded. Various examples show that these results are sharp.

Note

This work is part of the research program of the network ‘Analysis and operators’, contract HPRN-CT 2000 00116, funded by the European Commission.

Article information

Source
Ark. Mat., Volume 43, Number 2 (2005), 365-382.

Dates
Received: 8 January 2004
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898905

Digital Object Identifier
doi:10.1007/BF02384785

Mathematical Reviews number (MathSciNet)
MR2173957

Zentralblatt MATH identifier
1128.47039

Rights
2005 © Institut Mittag-Leffler

Citation

Esterle, Jean. Distance near the origin between elements of a strongly continuous semigroup. Ark. Mat. 43 (2005), no. 2, 365--382. doi:10.1007/BF02384785. https://projecteuclid.org/euclid.afm/1485898905


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