Spectral properties of general advection operators and weighted translation semigroups

We investigate the spectral properties of a class of weighted shift semigroups $(\mathcal{U}(t))_{t \geq 0}$ associated to abstract transport equations with a Lipschitz continuous vector field $\mathcal{F}$ and no--reentry boundary conditions. Generalizing the results of [25], we prove that the semigroup $(\mathcal{U}(t))_{t \geq 0}$ admits a canonical decomposition into three $C_0$-semigroups $(\mathcal{U}_1(t))_{t \geq 0}$, $(\mathcal{U}_2(t))_{t \geq 0}$ and $(\mathcal{U}_3(t))_{t \geq 0}$ with independent dynamics. A complete description of the spectra of the semigroups $(\mathcal{U}_i(t))_{t \geq 0}$ and their generators $\mathcal{T}_i$, $i=1,2$ is given. In particular, we prove that the spectrum of $\mathcal{T}_i$ is a left-half plane and that the Spectral Mapping Theorem holds: $\mathfrak{S}(\mathcal{U}_i(t))=\exp$ {$t \mathfrak{S}(\mathcal{T}_i)$}, $i=1,2$. Moreover, the semigroup $(\mathcal{U}_3(t))_{t \geq 0}$ extends to a $C_0$-group and its spectral properties are investigated by means of abstract results from positive semigroups theory. The properties of the flow associated to $\mathcal{F}$ are particularly relevant here and we investigate separately the cases of periodic and aperiodic flows. In particular, we show that, for periodic flow, the Spectral Mapping Theorem fails in general but $(\mathcal{U}_3(t))_{t \geq 0}$ and its generator $\mathcal{T}_3$ satisfy the so-called Annular Hull Theorem. We illustrate our results with various examples taken from collisionless kinetic theory.