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A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators
Author(s) -
Frosali Giovanni,
van der Mee Cornelis V. M.,
Mugelli Francesco
Publication year - 2004
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.495
Subject(s) - mathematics , semigroup , resolvent , characterization (materials science) , generator (circuit theory) , spectrum (functional analysis) , operator (biology) , pure mathematics , discrete mathematics , quantum mechanics , power (physics) , biochemistry , materials science , physics , chemistry , repressor , transcription factor , gene , nanotechnology
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.
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