On integrated semigroups which are not exponentially bounded (Q1320553)

Using the Laplace transform methods, the theory of exponentially bounded \(\alpha\)-times integrated semigroups was introduced and studied by H. Kellermann, W. Arendt and M. Hieber. In this article the authors treated the notion of \(\alpha\)-times integrated semigroups, which are not exponentially bounded on a Banach space \(X\). It should be noted that the Laplace transform techniques are not available in this case. Some properties of \(\alpha\)-times integrated semigroups and characterization of their generators are given. As an application, a bounded perturbation result of \(\alpha\)-times integrated semigroups and an adjoint theorem is proved.