Let be a (not necessarily exponentially bounded, not necessarily nondegenerate)-times integrated -semigroup and let be the generator of a group commuting with and . Under suitable conditions on and we prove the existence of an -times integrated -semigroup , which has generator provided that is nondegenerate and has generator . Explicit expressions of in terms of and are obtained. In particular, when is bounded, can be constructed by means of a series in terms of and powers of .
Yuan-Chuan LI. Sen-Yen SHAW. "Perturbation of non-exponentially-bounded -times integrated -semigroups." J. Math. Soc. Japan 55 (4) 1115 - 1136, October, 2003. https://doi.org/10.2969/jmsj/1191418767