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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
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