Arkiv för Matematik

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

Distance near the origin between elements of a strongly continuous semigroup

Jean Esterle

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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.


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

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

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

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2005 © Institut Mittag-Leffler


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.

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