Abel Ergodic Properties and Spectral Decomposition for Locally Integrable Semigroups

The question regarding the relationship between the spectral decomposition property of operators of a strongly continuous semigroup and that of its generator has been left open for many years. This paper proves that if one of the operators of a strongly continuous semigroup has the spectral decomposition property, then its generator does also. An example shows that the converse is not true in general.