Distance near the origin between elements of a strongly continuous semigroup

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 t→s(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.