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This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L 1 -space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C 0 -semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.

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