A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators
Giovanni Frosali, Cornelis V. M. van der Mee, Francesco Mugelli · Mathematical Methods in the Applied Sciences · 2004
Abstract 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 $G=\overline{A + B}$ is stated. The results are based on the spectral analysis of B(λ‐A)‐1. The main point is to study the conditions under which the value 1 belongs to the resolvent set, the continuous spectrum, or the residual spectrum of B(λ‐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. Copyright © 2004 John Wiley & Sons, Ltd.