Zohra BENDOUAD, Isabelle CHALENDAR, Jean ESTERLE, Jonathan R.PARTINGTON
This paper is concerned first with the behaviour of differences T(t)-T(s)near the origin,where(T(t))is a semigroup of operators on a Banach space,defined either on the positive real line ora sector in the right half-plane(in which case it is assumed analytic).For the non-quasinilpotent caseextensions of results in the published literature are provided,with best possible constants;in the case ofquasinilpotent semigroups on the half-plane,it is shown that,in general,di?erences such as T(t)?T(2t)have norm approaching 2 near the origin.The techniques given enable one to derive estimates of otherfunctions of the generator of the semigroup;in particular,conditions are given on the derivatives nearthe origin to guarantee that the semigroup generates a unital algebra and has bounded generator.