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

Abstract. We present a new method for constructing C0-semigroups for which properties of the resolvent of the generator and continuity properties in operator topology are controlled simultaneously. It allows us to show that a) there exists a C0-semigroup continuous in operator 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 continuous in operator 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 pos-sibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent decay. 1