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

✍ Scribed by Giovanni Frosali; Cornelis V. M. van der Mee; Francesco Mugelli

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 148 KB
• Volume: 27
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.495

✦ Synopsis

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 L^1^‐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.