Optimal individual stability estimates for C-0-semigroups in Banach spaces

In a previous paper we proved that the asymptotic behavior of a C-0-semigro up is completely determined by growth properties of the resolvent of its ge nerator and geometric properties of the underlying Banach space as describe d by its Fourier type. The given estimates turned out to be optimal. The me thod of proof uses complex interpolation theory and reflects the full semig roup structure. In the present paper we show that these uniform estimates h ave to be replaced by weaker ones, if individual initial value problems and local resolvents are considered because the full semigroup structure is la cking. In a different approach this problem has also been studied by Huang and van Neerven, and a part of our straightforward estimates can be inferre d from their results. We mainly stress upon the surprising fact that these estimates turn out to be optimal. Therefore it is not possible to obtain th e optimal uniform estimates mentioned above from individual ones. Concernin g Hardy-abscissas, individual orbits and their local resolvents behave as b adly as general vector valued functions and their Laplace-transforms. This is in strict contrast to the uniform situation of a C-0-semigroup itself an d the resolvent of its generator where a simple dichotomy holds true.