## A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory.(English)Zbl 0648.47015

The author proves the following theorem extending previous work by K. Jörgens [Commun. Pure Appl. Math. 11, 219-242 (1958; Zbl 0081.441)], I. Vidav [J. Math. Anal. Appl. 30, 264-279 (1970; Zbl 0195.137)] and J. Voigt [Monatshefte Math. 90, 153-161 (1980; Zbl 0433.47022), J. Math. Anal. Appl. 106, 140-153 (1985; Zbl 0567.45002)].
Let X be one of the Banach spaces $$L^ p(\Omega,\mu)$$ or C(M), M compact and metric or compact and extremely disconnected, $$S_ t$$ a strongly continuous semigroup in X with generator A. Let $$T_ t$$ be generated by $$A+K$$, where K is bounded and for some $$n\in {\mathbb{N}}$$ the operators $$KS_{s_ 1}KS_{s_ 2}...KS_{s_ n}K$$, $$s_ i>0$$, are strictly singular, i.e. no restriction to an infinite dimensional subspace induces an isomorphic embedding. Then $$r_ e(T_ t)=r_ e(S_ t)$$, $$t>0$$ where $$r_ e(\cdot)$$ denotes the essential spectral radius. From this some information on the spectral properties of the neutron transport equation is drawn.
Reviewer: A.Schröder

### MSC:

 47A55 Perturbation theory of linear operators 47D03 Groups and semigroups of linear operators