Operators L1(R+)→X and the norm continuity problem for semigroups

We present a new method for constructing C0-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 C0-semigroup which is continuous in the operator-norm topology for no t∈[0,1] such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; (b) there exists a C0-semigroup which is continuous in the operator-norm topology for no t∈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.