Semigroup realization of the transport--renewal operator

Verify the semigroup hypothesis in Assumption~\ref{ass:r0_semigroup} for the specific transport--renewal operator, including domain characterization, generation of the positive semigroup, and confirmation of the identity \(\mathcal K=-\mathcal F\mathcal A_0^{-1}\).

Background

The reduced threshold-sign theorem is conditional on a linear semigroup framework. The paper does not derive this framework from the PDE analysis and states that its verification for the particular transport--renewal operator requires identifying the operator domains, proving generation, and establishing the relation between the recruitment operator and the next-generation operator. Resolving this issue would make the reduced reproduction-number theory unconditional for the specified transport--renewal structure.