The operator $a(x)\frac {d}{dx}$ on Banach space

The operator $a(x)\frac {d}{dx}$ on $C(I)$, where $I$ is an interval contained in the real line, is considered in many places. In this paper, we attempt to reconsider it in the subspace of $C_0(-\infty ,\infty )$ containing all even functions, and show that it generates a strongly continuous semigroup. It is interesting that our main conditions seem contradictory to previous ones. It is due to the symmetry of the functions and the different domain of the operator than usual.