This is an announcement for the paper “Operators $L^1 (\mathbb R_+ )\to X$ and the norm continuity problem for semigroups” by R. Chill and Yu. Tomilov.
Abstract: We present a new method for constructing $C_0$-semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show that a) there exists a $C_0$--semigroup which is continuous in the operator-norm topology for $t\in [0,1]$ such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a $C_0$--semigroup which is continuous in the operator-norm topology for no $t\in R_+$ such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one. These examples rule out any possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent-norm decay on vertical lines.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1602.01163