Asymptotic behavior at the origin of the distance between elements of a strongly continuous semigroup
by
Jean Esterle
Universite de Bordeaux I

A semigroup T(t)_{t > 0} is said to be nontrivial if there exists t > 0 such that T(t) =/= 0. We show that if (T(t)_{t > 0} is a nontrivial strongly continuous semigroup of bounded operators on a Banach space X, and if there exists \tau > 0 and a positive continuous function f on [0, \tau] , with f(0) = 0 , satisfying \norm T(t) - T(t + f(t)) \norm < f(t)[(t^[ t/f(t)])/((t + f(t))^{1+ [ t/f(t)]})] for 0 < t <= \tau then the closed subalgebra A of B(X) generated by the semigroup (T(t))_{t > 0} is unital , and there exists S in A such that T(t) = e^tS for t > 0. This result is based on an inequality concerning quasinilpotent semigroups : if (T(t)_{t > 0} is a nontrivial quasinilpotent semigroup of bounded operators , then there exists \tau > 0 such that \norm T(t) - T(s) \norm > (s-t)[(t^[ t/(s-t)])/(s^[ s/(s-t)])] for 0 < t < s <= \tau.

Date received: June 23, 2002