The spectral mapping theorem for the exponential function with applications to the transport theory

The linear transport theory gave rise to new investigations of the spectral mapping theorem (SMTh) esup(sigma(-T)t) = sigma(esup(-Tt))minus(0) (for all t>0) for the exponential function esup(-Tt) generated by an unbounded operator -T. The original problem was to determine the asymptotic behaviour of the semigroup esup(-Tt), if one knows the spectrum of the generator -T. The best way to achieve this is to show that (SMTh) holds for the generator -T. (SMTh) and some of its weaker forms are investigated. Several of them are proved and there are also two new counterexamples to (SMTh) which illustrate a situation similar to that in the transport theory. Knowing the asymptotic part of the spectrum of the generator -T, one can determine the asymptotic behaviour of the semigroup esup(-Tt) which corresponds to the solution of the linear Boltzmann equation. A new condition on the scattering kernel in the Boltzmann equation was discovered which determines the existence of eigenvalues in the asymptotic part of the spectrum of the generator. This condition is far weaker than those ones already known. (Author)