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We present a new method for constructing $C_0$-semigroups for whichproperties of the resolvent of the generator and continuity properties of thesemigroup in the operator-norm topology are controlled simultaneously. Itallows us to show that a) there exists a $C_0$-semigroup which is continuous inthe operator-norm topology for no $t \in [0,1]$ such that the resolvent of itsgenerator has a logarithmic decay at infinity along vertical lines; b) thereexists a $C_0$-semigroup which is continuous in the operator-norm topology forno $t \in \mathbb R_+$ such that the resolvent of its generator has a decayalong vertical lines arbitrarily close to a logarithmic one. These examplesrule out any possibility of characterizing norm-continuity of semigroups onarbitrary Banach spaces in terms of resolvent-norm decay on vertical lines.

Operators $L^1 (\mathbb R_+ )\to X$ and the norm continuity problem for semigroups