In this paper, we study the existence of the flow of a hyperbolic system on some functional spaces.

Let $G$ be a Lie group, ${\mathcal G}$ its Lie algebra and $T^*G$ its cotangent bundle. On $T^*G,$ we consider the Lie group structure obtained by performing a left trivialization and endowing the resulting trivial bundle $G\times {\mathcal G}^*$ with the semi-direct product, using the co-adjoint action of $G$ on the dual space ${\mathcal G}^*$ of ${\mathcal G}$. We investigate the group of automorphisms of the Lie algebra ${\mathcal D}:=T^*{\mathcal G}$ of $T^*G.$ More precisely, we fully characterize the Lie algebra of all derivations of ${\mathcal D},$ exhibiting a finer decomposition into components made of well known spaces. Further, we specialize to the cases where $G$ has a bi-invariant Riemannian or pseudo-Riemannian metric, with the semi-simple and compact cases investigated as particular cases.

In this article, we consider dissipative Dirac system in the limit-circle case. Then using the Livsic's theorem, we prove the completeness of the system of root vectors for dissipative Dirac system with transmission conditions.