A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators

We consider a class of abstract evolution problems characterized by the sum of two unbounded linear operators A and B, where A is assumed to generate a positive semigroup of contractions on an L1-space and B is positive. We study the relations between the semigroup generator G and the operator A + B. A characterization theorem for G =A + B is stated. The results are based on the spectral analysis of -A)-1. The main point is to study the conditions under which the value 1 belongs to the resolvent - A)-1. Applications to the runaway problem in the kinetic theory of particle swarms and to the fragmentation problem describing polymer degradation are discussed in the light of the previous theory.