The paper is devoted to the spectral properties of a class of weighted shift semigroups arising in kinetic theory: \[ {U}(t)f({\mathbf x})=\exp\left[-\int_{0}^t\nu(\Phi({\mathbf x},-s))ds\right]f (\Phi({\mathbf x},-t)\chi_{\{t<\tau_-({\mathbf x})\}}({\mathbf x}), \] where the flow \(\Phi\) is associated to a globally Lipschitz transport field \({\mathcal F}\), that is, \(\Phi\) is the maximal solution of the Cauchy problem \({\mathbf X}'(t)={\mathcal F}({\mathbf X}(t)), \;{\mathbf X}(0)={\mathbf x}\in\Omega\subset{\mathbb R}^N,\) and \(\tau_\pm({\mathbf x})\) are stay times for \(\Phi({\mathbf x},\pm s)\) in \(\Omega\).
Spectral properties of the semigroup are investigated due to the canonical decomposition of \(\{{U}(t): t\geq0\}\) into three semigroups \(\{{U}_i(t): t\geq0\}\), \(i=1,2,3\), of class \(C_0\) with independent dynamics. It is proved that the spectra of \(U_1\) and \(U_2\) are in a left-half plane and the spectral mapping theorem for their generators holds. The semigroup \(U_3\) is proved to have properties similar to properties of a group. Properties of \(\Phi\) associated to \(\mathcal{F}\) are investigated separately for the cases of periodic and aperiodic flows. Some illustrative examples from kinetic theory are given.