Resolvent Norm Decay Does Not Characterize Norm Continuity

It is the fundamental principle of semigroup theory that the behavior of a strongly continuous semigroup (T (t))t≥0 on a Banach space X and the properties of its generator (A,D(A)), or equivalently the properties of the resolvent function R(λ,A) = (λ − A)−1 (λ ∈ C), should closely correlate. Indeed, the Laplace transform carries the regularity properties of the semigroup to the resolvent function while the several inversion and approximation formulas (TrotterKato, Post-Widder, etc.) allow the reconstruction of the semigroup from the resolvent (see [7], [17] or [18] for the relevant techniques and results in semigroup theory). And the correlation is indeed perfect if the participants are so: the analytic semigroups, the differentiable or merely eventually differentiable ∗ This research was carried out during the author’s stay at Arbeitsgruppe Funktionalanalysis of Universität Karlsruhe with the support of the Alexander von Humboldt-Stiftung and of the Gemeinnützige Hertie-Stiftung. The research was partially supported by the OTKA Grants F 43620, T 49786 and T 37758. Received May 11, 2006