Properties of a class of C0 semigroups on Banach spaces and their applications

In this paper we investigate some properties of a class of C0 semigroups on Banach spaces. Suppose that the spectrum of the infinitesimal generator is discrete and there are only finitely many eigenvalues in each vertical strip, we show that such semigroups can be expanded by their generalized eigenvectors under certain conditions. As a consequence, we assert that the semigroups with these conditions are eventually differentiable provided that the system of generalized eigenvectors is complete. As an example, we apply our results to a one-dimensional wave control system.