The time dependent transport equation with periodic boundary conditions is discussed in the space of continuous functions. Some aspects of the spectral properties of the corresponding transport operatorAandTF(t) (which is the strongly continuous semigroup generated byAin the subspaceF=D(A)) are studied, and it is shown that σ(TF(t)){λ:∣λ∣ exp (β1t)}, where β1is any real constant greater than the infimum of the total collision frequency, contains only finitely many elements, each of which is an eigenvalue ofTF(t) with finite algebraic multiplicity. Consequently, the asymptotic behavior of the time dependent solution is obtained.