Abstract
Set
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 functiont→s(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.
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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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Esterle, J. Distance near the origin between elements of a strongly continuous semigroup. Ark. Mat. 43, 365–382 (2005). https://doi.org/10.1007/BF02384785
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DOI: https://doi.org/10.1007/BF02384785